What is the meaning of the word "particle" in particle physics? I want to use Matt Strassler's definition of the word "particle" as a specific example:
Matt Strassler writes:

(1) "...all the elementary “particles” (i.e. quanta) of nature are
  quanta of waves in fields..."

My first question is, in this sentence, is the word "quanta" short for "quanta of waves in fields?"
Sentence (1) simply defines the word "particle" to be another name for "quanta".
I looked up quanta and quanta means

a wave that for a given frequency, have minimum energy and amplitude.

So, a quantum is a wave. Is this correct?
From this analysis, I conclude that, in particle physics, a particle is a wave, because a particle is defined as a quantum.
In mathematical terms:
$quantum \equiv wave$
or
$particle \equiv wave \equiv quantum$
So, in particle physics, the word particle, has the properties of a wave, and it has no other properties. A particle, in particle physics, for instance, is not a spherical object with finite radius.
This is confirmed by Matt Strassler, who writes 
(2) "...the word “particle” in particle physics has the same meaning as the word “quantum”..."
Sentence (2) is incomplete, because the word "particle" has several other meanings besides being another name for quanta. This is very confusing to non-physicists because the word "particle" has been defined many times and physicists pick and choose a meaning that saves the context.
My questions are:
1) The word "quantum" is a perfectly good word meaning a wave with minimum amplitude; why are physicists insisting on calling a quantum a particle?
2) Is there a unique definition of the word particle in physics not dependent on context (not casuistic) and that all physicists can agree upon?
If you cannot answer any of these, can you clarify if "particle" is another name for "quantum".
 A: 
My questions are:
1) The word "quantum" is a perfectly good word meaning a wave with minimum amplitude; why are physicists insisting on calling a quantum a particle?

The word "quantum" in physics is used as a "definite quantity of a variable" in contrast to "a continuous quantity of that variable". There are quanta of energy, for example the energy levels of an  atom which can be probed  by a quantized photon energy

Neutrons and protons can be considered as quanta building up a nucleus.
The location of atoms and molecules in a crystal can be considered as space quanta, because only specific sites can be accessed.
So the definition of a "quantum" as a "wave" is wrong. A particle may be a quantum if it builds up a specific unit, as for example the quarks can be considered as quanta which build up hadrons ( bricks to be crude).
2) Is there a unique definition of the word particle in physics not dependent on context (not casuistic) and that all physicists can agree upon?
Yes, a particle  is an entity whose position can be defined  by an (x,y,z,t) and its four momentum by ( p_x,p_y,p_z,E) as with macroscopic particles. That is why it is called a particle. BUT talking of elementary particles quantum mechanics reigns and  the Heisenberg Uncertainty Principle holds.
In  quantum mechanics, what can be measured as a particle in one experiment as here we see an antiproton hitting a proton at rest and generating a number of pions,

where we can measure the four momentum of each particle in the picture and the track it left in space, can be a probability wave in another experiment. 
For completeness here is the image used in a similar  question:

This two slit experiment with an electron beam starts with individual electrons, (particles)  and detectors are counting the location of their impact. In a) it looks random  and it is only after enough statistics is accumulated that the interference pattern is observed, on the  distribution in the x y plane  of individual hits. A materialization of the probability distribution. In this particular experiment the slit through which the individual electron particle) passed  could be identified, non destructively. The interference pattern is there. The boundary condition of two slits is what shows the probability wave nature of the particle.
A: There are two definitions of particle in physics, both commonly used. I prefer the following terms:


*

*Asymptotic particle: The fundamental incoming S-matrix states.

*Field particle: an excitation of a fundamental field in a physical Lagrangian.


The two definitions are different in the case of composite particles, which are particles only in the sense of 1 (except for that they are also type 2 particles of effective theories) and for particles like quarks and gluons, which are only particles in the second sense but not the first.
The first definition is Wigner's, and it dominated high energy physics until 1974. Once quarks were recognized to be real and confined, the second definition took over. The two definitions are identical in free field theories, and they overlap in interacting field theories with nonconfined particles. But for strong interactions, the spectrum particles are hadrons, while the fundamental fields are quarks and gluons, they are completely different, and require different tools to analyze.
The spectrum particles in the strong interactions are described by S-matrix theory. They are approximately like strings in string theory. The fundamental particles are described by field theoretic perturbation theory. Much of physics of the last 30 years is devoted to reconciling the two different kinds of descriptions.
A: In his article http://profmattstrassler.com/articles-and-posts/particle-physics-basics/fields-and-their-particles-with-math/5-waves-quantum/, Strassler tries to present an explanation of particles in quantum field theory (QFT) in the most elementary terms possible. The result is that he draws a picture and uses terminology that is not commonly used, though it gives some insight into what QFT is about. 
So let me write a narrative in slightly more abstract terms, but in the standard notation and terminology, and then relate it to what Strassler writes.
Just like the waves of a single oscillator (think of a vibrating string) can be represented as a superposition of harmonic waves with given frequencies $\omega$ (Fourier analysis), so the waves of an extended medium (such as water) can be represented as a superposition of plane waves with given 4-momentum vectors $p$. The vectors $p$ label the different modes of the superposition 
$p$ is composed of $p_0=\pm E/c=\pm \omega \hbar/c$ where $E$ is the energy, $\omega$ the angular frequency (and $\hbar/c$ is a constant proportionality factor) and a 3-vector $\mathbb p$, the spatial momentum, telling the direction and speed of the motion of the wave. They are related by the so-called dispersion relation $p_0^2= \mathbb p^2+(mc)^2$, where $m$ is the mass of the (free) field. If there is just a single mode (rather than a superposition, which is the usual case) then an observer moving with the speed of the wave will see the 4-vector with zero spacial momentum (rest frame); so one can consider it to have just one component determined by the frequency.
Now a quantum field is not an ordinary field but an operator-valued field.
You may think of an operator as a random variable, which takes different values - called eigenvalues - upon each measurement (if it can be measured) according to a probability distribution characteristic of the state of the field. (Though the measurement of fields is a complex subject, so this is quite simplified.)
Doing the decomposition into plane waves with a quantum field, each amplitude becomes an operator $a(p)$ or $a(\omega)$ itself, called a creation or annihilation operator depending on the sign of $p_0$; the creation operators have an additional * in the formula. These operators have arbitrary complex eigenvalues (just as in a classical field, the amplitudes are arbitrary comlex numbers). [Thus there is no smallest amplitude.] 
However, due to the peculiarities of eigenvalues (which one can see already when the operators are just 2 by 2 matrices), the eigenvalues of a product are generally not equal to the products of the eigenvalues of the factors. In the case of a single quantum oscillator, it turns out that the product $a^*a$ has very few eigenvalues only, namely only the nonnegative integers. If you would measure $a^*a$ (which is not actually possible), you would therefore get one of the numbers $n=0,1,2,...$, counting the excitation level (or the ''number of quanta'', though this terminology is rarely used). One therefore calls $a^*a$ the number operator. Level 0 is the ground state, and higher and ligher levels have more and more energy.
Now we take all modes into account, allowing for the full field in place of just one plane wave mode. Then the $a$ depend on the 4-momentum $p$, and to get
a number operator  one must integrate the operators $a^*(p)a(p)$ over all possible 3-momenta. Then again, the eigenvalues of the number operator are nonnegative integers $n$, this time interpreted as the number of particles detected. The standard way of talking about this situation is by saying that, in QFT, ''a particle is an elementary excitation of a quantum field''. 
Identifying an elementary excitations with a quantum, a single particle becomes a single field quantum. Moreover, if one simplify the noncommutative aspects of operators by treating the amplitude $a$ as a kind of square root of the number $n$, one might say that a quantum is ''a wave with minimal aplitude'', though this is quite misleading if you attach more than metaphorical meaning to it.
This gives Strassler's imagery.
Note that in a general state of a quantum field, it is meaningless to talk about the number of particles it contains, just as it is meaningless to talk about the position of a quantum oscillator. What one can talk about is about the mean position of the oscillator and the mean number of particles, which is usually enough to draw macroscopic conclusion (in quantum statistical mechanics). 
Particle detection events, on the other hand, concern very special (asymptotic) states, where interaction with recording equipment singles out a particular measured state with a definite observable particle content. It is this particle content that is represented by the traces in bubble chambers and photographs from CERN such as those mentioned in the answer by anna_v.
A: A particle is the name for a quantum system for which $P_\mu P^\mu$ is a positive multiple of the identity (the mass squared), the energy is positive, and the angular momentum takes only certain discrete values (spin).
EDIT: I will add below a few extra comments because the original question was about thee relation of particles and waves.
A field $\phi(x)$ associated to a certain particle is an operator (or a collection of operators, one for every spacetime point) that sends the vacuum state $|0\rangle$ into a particle localized in space and time, $\phi(x)|0\rangle=|\phi(x)\rangle \approx |x\rangle$. It turns out that for consistency of the theory (which is a QFT) these operators $\phi(x)$ satisfy certain Lorentz invariant wave equations. This in particular mean that $|\phi(x)\rangle$ and the various amplitudes $\langle A|\phi(x)\rangle$ obey those wave equations. 
A: Assuming that "particle" refers to elementary particles, then yes. An elementary particle is a quantum of energy. 
Basically the confusion arises by people trying to simplify what is really a subtle distinction, and adding to that there has historically been some confusing terminology. 
Note - Particle in context of my post will always refer to an elementary particle
No doubt you have heard of the uncertainty principle. What that says is that the position and momentum of a particle cannot be simultaneously known to some arbitrary precision. There is an uncertainty associated with the particle's position and momentum, which is fundamental in nature. It is inherently 'fuzzy' in some sense, with it's position not perfectly defined. 
But how exactly can we say where a particle is in the first place? Or tell how fast it's moving? That is given by the particle's wavefunction. Now even though the name suggests that it is a wave, what the wavefunction actually describes is the probability that the particle is at a given location at a give time. For a 'free particle' (which is a particle with no external forces acting on it), the wavefunction is defined by a 'wavepacket'. This wikipedia page has a nice diagram explaining that. 
The wavepacket has a reasonably well defined position and momentum at any given point, therefore we can pinpoint where it is quite well. This is what people refer to as particles - the quantized wavepacket, which is actually just a wavefunction that describes a probability. Therefore what all of the above is saying is that when you can predict reasonably accurately where something is, and how fast it's going, we can call that a particle. Which isn't a bad definition at all. :)
A: Contrary to what Matt Strassler writes in his blog, there no waves in particle physics and this is why the discipline is named "particle physics" and not "wave physics".
Intuitively an elementary particle is the most basic piece of matter known. Each elementary particle is characterized by a set of physical properties: mass, charge, and spin. E.g. a photon is an elementary particle with zero mass, zero charge and spin $1$. Everything around us is made of lots of those elementary pieces. A good introduction to those themes is given in the CERN site:
http://public.web.cern.ch/public/en/science/Glossary-en.php#P
http://public.web.cern.ch/public/en/science/standardmodel-en.html
A discussion of why a particle is not a wave neither has the properties of a wave is given here
http://statintquant.net/siq/siqse3.html#x42-60003
Let us see the rest of questions.
No a quantum is not a wave. The term quantum usually refers to a small --of the order of Planck size-- quantity of a physical property. A quantum of energy is a small quantity of energy: e.g., the small difference of energy between two levels of energy in a hydrogen atom. E.g. a chronon is the hypothetical quantum of time.
Elementary particles are quantum particles and quantum particles are not spherical objects with finite radius. In fact technically elementary particles are considered pointlike objects
About your specific questions:
1) As stated "quantum" does not mean "a wave with minimum amplitude".
2) Yes, technically an elementary particle is defined as an irreducible element of the Poincaré group. In more laymen terms this means that any elementary particle is unambiguously identified and characterized by its mass, charge, and spin.
A: I will try to add my 50 cents to the discussion. My understanding is that when we speak of particles in quantum context we are essentially transferring our intuitive classical concepts to the quantum realm.
What do we understand we speak of particles in normal life? We think of certain integrate entities more of less local in space that make up a larger system (${\it part}$-icle) while still keeping some sort of identity. As a consequence of the latter, the expectation is that all extensive macroscopic properties of the total system scale linearly with the number of particles.
When dealing with QFT it is often the case that in the low-energy limit the system can be reduced a set of harmonic oscillators (let's limit it to the bosonic fields for now). What is the most important feature of harmonic oscillator spectrum? It is equidistant. The energy of the oscillator is linearly proportional to the excitation number, and it always costs the same amount of energy to increase the excitation state by one step. Superficially this looks very similar to what one would expect when dealing with classical non-interacting particles (same energy per particle; total energy proportional to the number of particles). Hence it is very tempting to identify the excitation degree $N$ of the certain degree of freedom ("number of quanta") with the number of "particles" in the corresponding "box" in the phase space.
Of course the similarity is not a mere coincidence, and the analogy is very far-reaching. But at the end of the day it is more of a habit when we think of quanta as particles. In fact the identification of quanta with common sense particles might at times become quite strenuous as is the case of photons (see e.g. https://link.springer.com/article/10.1007/BF01135846).
