What is enthalpy? After an intense research reading documents and Q&A, I am still very confused about the concept Enthalpy.
Etymology says: "to warm in". [ἐνθάλπω (enthálpō, “to warm in”)]
I interpret: "to put warmth (same as heat) into something." (i. e. warmth transfer = heat?)
[1] Some will say: "Enthalpy is a measure of heat transfer".
I will wonder: "Being heat[-flow] already a concept involving transference, what's the point of defining a concept that refers to a transfer-transference?"
[2] Some will say: "Enthalpy is a state function to correct the fact that heat is not a state function".
I will wonder: "Okay, but how and why?".
[3] Some will say: "Enthalpy is Internal energy plus Pressure times Volume".
I will wonder: "Looks just like a brother-concept to what I learned as heat. Still I see no point in it.".
I would like to understand not only what enthalpy is, but where can I visualize enthalpy in for example a cup of coffee that is chilling on a balcony, in contact with the atmosphere.
 A: One the main advantages of enthalpy is that it allows you to work out compression and expansion work done during constant pressure thermodynamic processes in an easier form.
Rather than think of the energy content of a system, we could include the work done to make room for the system in the first place.
You need to do work to create space for a system, and this work can be estimated using PV, which is the volume occupied by the system multiplied by the pressure of the environment in which the system is to be created.
So taking U as the total internal energy of a system, then we can define H = PV plus U.
So H is the total energy required to construct the system and also make room for it.
To reverse the idea, if you completely destroy a system, you would recover not only the internal energy of the system, as well as the work done by the environment (atmosphere), as it rushes in to fill the vacuum.
There are only two causes for an increase in the enthalpy of any particular  system, either the system expands and it does work on the environment to create space for this expansion, or the internal energy of the system increases.
We can say $\Delta H  = Q  +  W_0$,  where $W_0$ is any other type of work and Q is the amount oF heat added to the system.

Where can I visualize enthalpy in for example a cup of coffee that is chilling on a balcony, in contact with the atmosphere

Enthalpy is a measure of the work done to move air out of the way to give space to your coffee cup (PV) plus the internal energy (U) of the hot coffee  inside the cup.
To sum up, the enthalpy change is produced solely by work, in various forms,  and heat. So this eliminates the calculations involved in compression expansion themodynamic processes. 
$\Delta H $ in a direct indication of the heat added to the system, as long as no other types of work is being done.
A: I find it easiest to think about enthalpy in the context of temperature.
Enthalpy is the amount of heat energy contained within an arbitrarily defined system. So for example, you define the coffee in the cup to be your system, enthalpy is the state variable that you can use to characterize the amount of heat in that cup. 
Take note that heat in the thermodynamic sense means the sum of the kinetic energies of each molecule in the system. So KE = 1/2 * m * v^2. That's why the diffusion of gasses occurs faster in higher temperature systems. Also, a reaction involving the The molecules are literally moving faster.
*It also includes the vibrational and orbital energies of every molecule in the system, but my personal view is that it is easier to just consider the kinetic energies. Because it is a better mental image to picture a bunch of molecules flying around, rather than a bunch of molecules shivering next to each other. 
Also notice the words state variable. Enthalpy is a state variable. The enthalpy in a system is constant until that system is acted upon by an external actor. Because it is a state variable, it would be very hard for us to exactly determine the enthalpy of a system without some known reference. That is why, when talking about a chemical reaction or system of reactions, you will often see the change in enthalpy as opposed to just enthalpy. It is easier to measure the change in state rather than the instantaneous state. 
But back to my original thought. The temperature of an object is best thought of as the energy difference that drives heat energy to move from one object to another until they reach equilibrium. While the enthalpy of each object is the actual amount of heat that is in each object. An analogy is a pressurized chamber that can be opened to the atmosphere. The chamber is at a higher pressure because it has a different set of state variables than the atmosphere (volume, quantity of gas [moles], temperature). And when it is opened up, the pressure will change until it reaches equilibrium with the atmosphere.
Take a look at the following picture and think of the enthalpy vs. temperature of each box.
 
