How to "read" the temperature of an abstract system? How can I interpret the parameter temperature $T$, if I'm not given the description of the system in terms of the equation of state, $E(S,V\ )$ or $S(E,V\ )$ and so on.
In many systems it makes sense to think of it as an "energy" itself, e.g. when the entropy is such, that $T$ basically represents the mean kinetic energy of particles. The general definition goes like 
$$\frac 1 T=\small{\left(\frac{\partial S(E,V\ )}{\partial E}\right)_V}.$$
However, very often, especially when I'm reading about phase transitions, examples for systems with critical exponents and so on, they usually talk about the parameter $T$ and an associated critical $T_c$ without taking any specific system into accound. There is usually some abstract free energy $F$, which pops out abstract quantities. Or in the Boltzman distribution and derived quantities, often something gets activated when the value of $kT$ catches up with some system specific energy value $E_0$. And it gets more complicated when it takes QFT like form. 
What is the temperature in a general setting. 
How do I read this kind of things and what should I have in mind when reading these kinds of texts? 
 A: The most fundamental definition of temperature is derived from the zeroth law of thermodynamics. 
The zeroth law declares thermal equilibrium an equivalence relationship, and thus we can tag each equivalence class with a number that we call temperature. Or in less mathematical term, temperature is a physical quantity tagged to each thermodynamic system such that any two systems with the same temperature would stay in thermal equilibrium when they contact.
The exact way of assigning temperature to a system is called a temperature scale. There were multiple scales before, most based on thermal properties of a particular substance. Then Kelvin devised a scale based solely on thermodynamic principles, which we call "absolute scale".
A: Stat Mech is all about taking averages with the proper measure.
The simplest measure is the micro-canonical measure (ensemble), in which you assume that the system has a given energy, and that all the states in this energy shell are equally probable. The assumption that the energy is known basically mean that the system does not exchange energy with a bath.
A much more useful measure is the canonical measure, in which you assume that the average energy is known, but that the system may exchange energy with a heat bath. It can be shown (and it is shown here) that if you assume that in each energy shell the states are equi-probable, this probability is proportional to $e^{-E/k_bT}$. 
This shows you that $T$ measures the weight that is given to states, according to their energy. When $T$ is high, the weight of a state with a given energy becomes higher, and when $T$ is low it becomes lower. It is easily seen that for $T\to\infty$ all the states become equi-probable and for $T\to 0$ only the ground state(s) is (are) counted.
You might ask then, "so why isn't the system almost always at the ground state? It is the most probable one!". Here entropy goes into play. By definition of the entropy $S(E)$, the number of states with an energy $E$ is $e^{S(E)/k_b}$, and it is a rapidly increasing function of $E$. Therefore, when you average over energies you should use the measure
$$e^{-\beta E}e^{S(E)/k_b}=e^{-\beta(E-TS)}$$
because higher energies have a much larger number of states on the shell. This is what I meant when I said that $T$ is a measure of "how much energy can I pay in order to buy some entropy" - high-entropy states are more probable from state-counting considerations, but are less probable from energetic considerations. $T$ is a measure of the relative balance of these two.
Also, your suggestion of functionally defining $T$ as "the thing that is equal for two systems in thermal contact" is great, and is actually used it in this textbook which gives offers a thorough, insightful introduction to "what is temperature". I highly recommend it for beginners.
A: Your definition is the thermodyanmics definition of temperature. Most discussions of phase transitions are coming from a statisical mechanics point of view. In this paradigm, the definition of temperature follows from the definition of entropy $S = k_b \log \Omega(E)$
$$ \frac{1}{kT} = \frac{d \log \Omega(E)}{dE}$$
Where $\Omega(E)$ is the number of microstates with energy $E$.
Wikipedia has a nice derivation of the connection between these two approaches.
