What is temperature in the field of classical thermodynamics?

What is temperature in the field of classical thermodynamics? In many (or all?) encyclopaedias, physics or thermodynamics books, the internet…the temperature T is described as a measure for the mean kinetic translational energy (= centre of gravity energy) E of an ideal gas. This does not say anything about the nature of temperature. The length of a piece of wire can also be a measure of temperature. It is given in some literature as T = 2/3*E in a coherent system of units for an ideal gas.
In my opinion, the temperature must not be a measure for the mean kinetic translational energy E, but it must be identical with it. Moreover, this energy or temperature must be related to a particle (or a quantity of particles), in contrast to mechanics, where energy is related to a mass.
As an illustrative example, we can mention air: here we have one-, two-, three-, … -atomic particles in the gas mixture, all of which have one property in common, namely the mean kinetic translational energy, and this energy must be the temperature.

My question: Can anyone tell me if this simple physical explanation or definition already exists somewhere or if there is a historical explanation (I have not found a hint in Boltzmann, Clausius or elsewhere.)?