What really is a wave and how it is treated mathematically? First of all, I know there's a much alike question here but this is not duplicate since I couldn't find there the answer I'm seeking. My problem is the following: I know that intuitively we have a wave when we have some quantitiy (that as I see can be anything) oscilating at each point in space. So for instance, electromagnetic waves are composed of electric and magnetic fields oscilating on each point of space.
Now, this is vague and imprecise. It is not clear at first how to model this mathematically and what properties this thing should have. However there's the wave equation:
$$\nabla^2 \psi = \dfrac{1}{v^2} \dfrac{\partial ^2}{\partial t^2}\psi,$$
but asking some physicists they told me that not every wave obeys that equation. That anything obeying that equation is really a wave, but that there are waves which evolve differently, some that are nonlinear and all of that.
In that case it seems everything is totally vague. A wave is something that moves like a wave, some of those things obeys a certain equation and the others can obey equations totally different. In that case it becomes a little bit difficult to grasp what really is a wave and how do we treat waves with precision.
So, what is a wave and how waves are preciselly dealt with in some mathematical framework?
 A: 
What really is a wave

A propagating disturbance in a material medium, e.g., air, or immaterial 'medium', e.g., the electromagnetic field.
A wave function is a mathematical description of the propagating disturbance and is a solution to some partial differential equation involving spatial and time derivatives.
One can quite simply construct a wave function by taking an ordinary function of one variable, e.g.
$$f(\theta) = \cos (\theta)$$
and replacing the argument with a function of the space and time coordinates, e.g.
$$\theta = \vec k \cdot \vec x - \omega t$$
so that the wave function is
$$f(x,t) = \cos(\vec k \cdot \vec x - \omega t)$$
This particular wave function is a solution to the wave equation in your question if
$$\frac{\omega^2}{k^2} = v^2$$
and is a sinusoid that propagates in the $\vec k$ direction with a phase velocity of $v$.

(From the Wikipedia article "Wave")
But, of course, there are other wave equations that some wave functions solve.  A somewhat famous one is
$$-\frac{\hbar^2}{2m}\nabla^2\psi + V(x)\psi  = i\hbar\frac{\partial}{\partial t}\psi$$
And it's still not clear precisely what or where the 'medium' is for these waves. 
A: First I would like to give you a classical standard definition of a wave. But at the same try to keep in mind that there are different types of waves. Some are transverse waves such as electromagnetic waves and some are longitudinal or pressure waves such as sound waves. These different types might create the vagueness you describe. EM waves travel through a vacuum and sound waves and water waves do not as they require a medium. Now the Classical Wave definition to build from follows.
In the classic sense a wave is a disturbance that travels through a medium. It transports energy from its source to another destination while at the same time not transporting matter from the source to the destination. Each individual particle of the medium is temporarily displaced and then returns to its original location. And each particle transfers the energy disturbance to its adjacent particle in the same direction thereby propagating the wave. 
The wave equations follow :
Mathematically, the most basic wave is the (spatially) one-dimensional sine wave (or harmonic wave or sinusoid) with an amplitude $u$ described by the equation:
$u(x,t)= $A$\sin$(kx-$\omega$t + $\phi$) , 
where
A is the maximum amplitude of the wave, maximum distance from the highest point of the disturbance in the medium (the crest) to the equilibrium point during one wave cycle. In the illustration to the right, this is the maximum vertical distance between the baseline and the wave.
$x$ is the space coordinate
$t$ is the time coordinate
$k$ is the wavenumber
$\omega$ is the angular frequency
$\phi$ is the phase constant.
The units of the amplitude depend on the type of wave. Transverse mechanical waves (e.g., a wave on a string) have an amplitude expressed as a distance (e.g., meters), longitudinal mechanical waves (e.g., sound waves) use units of pressure (e.g., pascals), and electromagnetic waves (a form of transverse vacuum wave) express the amplitude in terms of its electric field (e.g., volts/meter).

Sinusoidal waves correspond to simple harmonic motion.
The wavelength $\lambda$ is the distance between two sequential crests or troughs (or other equivalent points), generally is measured in meters. A wavenumber $k$, the spatial frequency of the wave in radians per unit distance (typically per meter), can be associated with the wavelength by the relation
$k=\frac{2 \pi}{\lambda}$ ,
The period T is the time for one complete cycle of an oscillation of a wave. The frequency f is the number of periods per unit time (per second) and is typically measured in hertz. These are related by:
$f=\frac{1}{T}$ ,
In other words, the frequency and period of a wave are reciprocals.
The angular frequency $\omega$ represents the frequency in radians per second. It is related to the frequency or period by
$\omega = 2 \pi f = \frac{2 \pi}{T}$ ,
The wavelength $\lambda$ of a sinusoidal waveform traveling at constant speed $v$ is given by:
$\lambda = \frac{v}{f}$,
As mention this equation satisfies a wave in one dimensional space.
For Examples :

About 10 years later a more complete wave equation emerged considering multidimensional space.
The wave equation is a hyperbolic partial differential equation. It typically concerns a time variable $t$, one or more spatial variables $x1, x2, …, xn$, and a scalar function $u = u (x1, x2, …, xn; t)$, whose values could model, for example, the mechanical displacement of a wave. The wave equation for u is
${ \partial^2 u \over \partial t^2 } = c^2 \nabla^2 u$ 
where $∇2$ is the (spatial) Laplacian and $c$ is a fixed constant.
Solutions of this equation describe propagation of disturbances out from the region at a fixed speed in one or in all spatial directions, as do physical waves from plane or localized sources; the constant c is identified with the propagation speed of the wave. This equation is linear. Therefore, the sum of any two solutions is again a solution: in physics this property is called the superposition principle.
Multispatial Example :

EM Waves :
The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field $E$ or the magnetic field $B$, takes the form:
$(c^2\nabla^2 - \frac{\partial^2}{\partial t^2})$ $\mathbf{E}$ = $\mathbf{0}$ 
$(c^2\nabla^2 - \frac{\partial^2}{\partial t^2})$ $\mathbf{B}$ = $\mathbf{0}$
where
$c = \frac{1}{\sqrt {\mu_0\varepsilon_0}}$ 
is the speed of light in a medium with permeability (μ0), and permittivity (ε0), and ∇2 is the Laplace operator. In a vacuum, c = 299,792,458 meters per second, which is the speed of light in free space.1 The electromagnetic wave equation derives from Maxwell's equations.
Now that I have given you a little background on waves and pointed out especially that their are different types of waves, EM, Sound, those that require a medium those and those that don't. And noting that some propagate from a point source in one direction while a pressure wave could propagate in all directions. And the noting that the cause of waves can be due to different types of disturbances it should be a little easier to imagine one equation will not work for all types, propagation sources, and different dimensional propagations. It should also be noted that anumber of other factors would impact the math as well, boundary conditions comes to mind.
Also another important class of problems occurs in enclosed spaces specified by boundary conditions, for which the solutions represent standing waves, or harmonics, analogous to the harmonics of musical instruments.
The wave equation, and modifications of it, are also found in elasticity, quantum mechanics, plasma physics and general relativity.  
So to get a handle on which mathematical model you need to work with you need to first
pick what wave you want to discuss and what and how the disturbance was sourced and into how many spatial dimensions, does the medium need to be considered or is it traveling in a vacuum and will a classical treatment suffice or does general relativity suffice or perhaps we need to use quantum mechanics to drive the solution home. 
An Example of a Boundary Condition like a drum. 

A: Julia, I note your comments.

What really is a wave?

It's an energetic dynamical deformation or displacement propagating through some medium. An ocean wave is like this, but don't think of it as just some up an down thing. See this image, imagine the waves are 1m high:
 GNUFDL image by Kraaiennest, see Wikipedia
Note the red test particles going round and round? They don't just go up and down, they exhibit angular momentum, and they don't stop circulating 1m down into the water. The wave isn't localized, instead it takes "many paths".   

and how it is treated mathematically?

In various ways. For example take a look at the travelling breather. It's treated differently because it's a different wave:

GFDL image by Georgiev DD, Papaioanou SN, Glazebrook JF. Neuronic system inside neurons: molecular biology and biophysics of neuronal microtubules. Biomedical Reviews 2004; 15: 67-75. DOI: 10.14748/bmr.v15.103a 

I know that intuitively we have a wave when we have some quantity (that as I see can be anything) oscillating at each point in space. 

Yes, we have a wave when something is waving.  

So for instance, electromagnetic waves are composed of electric and magnetic fields oscillating on each point of space.

I'm afraid that's not true. The wave concerned if the electromagnetic wave. And if you could ask Maxwell, he would say “light consists of transverse undulations in the same medium that is the cause of electric and magnetic phenomena". If you could press him about what's actually undulating, I think he would say space itself. This maybe sounds unfamiliar, but check out LIGO: Gravitational waves are ripples in space-time (the fabled “fabric” of the Universe). When an ocean wave moves through the sea, the sea waves. When a seismic wave moves through the ground, the ground waves. And when an electromagnetic wave moves through space...   

Now, this is vague and imprecise. It is not clear at first how to model this mathematically and what properties this thing should have. However there's the wave equation: $\nabla^2 \psi = \dfrac{1}{v^2} \dfrac{\partial ^2}{\partial t^2}\psi$ but asking some physicists they told me that not every wave obeys that equation. That anything obeying that equation is really a wave, but that there are waves which evolve differently, some that are nonlinear and all of that. In that case it seems everything is totally vague. A wave is something that moves like a wave, some of those things obeys a certain equation and the others can obey equations totally different. In that case it becomes a little bit difficult to grasp what really is a wave and how do we treat waves with precision.

I think you're coming at this from the wrong end. A wave is what it is, whether it's an ocean wave, a sound wave, a seismic P wave or S wave, a gravitational wave, an electromagnetic wave, or some other kind of wave. These waves don't "obey" some equation. Instead the equation attempts to model the wave in a rigorous way. 

So, what is a wave and how waves are precisely dealt with in some mathematical framework?

As above, and they're precisely dealt with in various ways because there's various waves. Sorry if that doesn't quite satisfy, but I don't know what else I can say. 

In particular a general definition of a wave should include the terminology of a "standing wave" as a special case and also of damped, nonlinear waves and shock waves.

Like I was saying, there's various waves. Note though that a standing wave isn't really motionless. You typically have the wave moving left and right at the same time, such that the oscillations don't appear to move, and the wave looks motionless. But it isn't. If you have a standing-wave photon in a cavity, which we think of as a gedanken mirror-box, then when you drop one of the sides it's off like a shot at c from a standing start. Because it was always moving at c.  

If one defines a wave as a solution to the wave equation above, a standing wave would be a wave, but it doesn't include nonlinear waves. So one could say: "a wave is a solution to a wave equation".

It isn't. A wave is a wave. It isn't the solution to an equation. Instead it's something that can knock you over in the surf. Or something that can level a city. Or something  from which you can make matter. An equation is just some symbols on a piece of paper.  

But what is a "wave equation"?

Something that describes a mathematical model of a wave. 

If one says something like "an organized propagating of imbalance" (as in researchgate.net/profile/John_Scales/publication/…) one misses the "standing waves". 

Reiterating what I said above, it's difficult to describe all the various wave types. Heck, people can't even tell you what the E=hc/λ photon actually is. Maybe you need to ask a less general question, about the photon?   
