I'm currently reading Fermi's "Thermodynamics" and I'm trying to grasp the (possibly different) right definitions for temperature, thermal equilibrium, heat.

Let's start with the latter. We can define heat through the 1° principle of Thermodynamics:$$Q=\Delta U + L ,$$that is, “Heat is the quantity of energy that a system absorbs from the ambient in a form that's not mechanical work.”. OK, I see no problem with this, apart from the fact that we need to define the energy $U$ of a thermodynamical system. Let's ignore it.

Now thermal equilibrium. This is the one I'm finding more troubles with, because in all definitions I've come across (maybe not very good ones, or maybe it's my interpretation) there's some reference to temperature, while in the definition of the latter there's reference to the concept of thermal equilibrium, but one has to start somewhere. For example, from (IT) Wikipedia, I read:

Thermal equilibrium: there is no flux of heat, temperature is constant in time and is the same in every point of the system.

The way that temperature is defined in the book is, first of all, the empirical (operational) one:

Temperature can be measured by putting a thermometer in contact with the system, for a sufficient time interval so that thermal equilibrium is established.

Some pages later there's also mention to the gas-thermometer. Finally, in the “Second principle of Thermodynamics” chapter, it's said:

Until now, we've only made use of an empirical scale of temperature. [...]If we put in thermal contact two bodies at a different temperature, heat will flow spontaneously by conduction from one body to the other. Now, by definition, we will say that the body from which the heat flows is the one with the higher temperature.

Now, clearly the definition in the first blockquote requires thermal equilibrium (beetween a body itself and beetween to bodies, I suppose) to be independently defined. Regarding the second, how can one tell what's the direction of the heat flux? Also, the second definition doesn't give a method to measure temperature, but only a way to tell which body is the hotter, right?

As I put them above, those definitions seem to me random pieces of a puzzle, I need to get a clearer picture. So any help is appreciated.

I'm currently reading Fermi's "Thermodynamics" and I'm trying to grasp the (possibly different) right definitions for temperature, thermal equilibrium, heat.

To clarify, I'm looking for definitions from a purely thermodynamical point of view, which is also the line followed by the book.

Let's start with the latter. We can define heat through the 1° principle of Thermodynamics:$$Q=\Delta U + L ,$$that is, “Heat is the quantity of energy that a system absorbs from the ambient in a form that's not mechanical work.”. OK, I see no problem with this, apart from the fact that we need to define the energy $U$ of a thermodynamical system. Let's ignore it.

Now thermal equilibrium. This is the one I'm finding more troubles with, because in all definitions I've come across (maybe not very good ones, or maybe it's my interpretation) there's some reference to temperature, while in the definition of the latter there's reference to the concept of thermal equilibrium, but one has to start somewhere. For example, from (IT) Wikipedia, I read:

Thermal equilibrium: there is no flux of heat, temperature is constant in time and is the same in every point of the system.

The way that temperature is defined in the book is, first of all, the empirical (operational) one:

Temperature can be measured by putting a thermometer in contact with the system, for a sufficient time interval so that thermal equilibrium is established.

Some pages later there's also mention to the gas-thermometer. Finally, in the “Second principle of Thermodynamics” chapter, it's said:

Until now, we've only made use of an empirical scale of temperature. [...]If we put in thermal contact two bodies at a different temperature, heat will flow spontaneously by conduction from one body to the other. Now, by definition, we will say that the body from which the heat flows is the one with the higher temperature.

Now, clearly the definition in the first blockquote requires thermal equilibrium (beetween a body itself and beetween to bodies, I suppose) to be independently defined. Regarding the second, how can one tell what's the direction of the heat flux? Also, the second definition doesn't give a method to measure temperature, but only a way to tell which body is the hotter, right?

As I put them above, those definitions seem to me random pieces of a puzzle, I need to get a clearer picture. So any help is appreciated.