While it is convenient to describe temperature as the average speed of molecules, it is really measuring the energy content. In solids this may affect the vibration rate of the molecules rather than movement from place to place and may also include internal movement of atoms within a molecule. As these become excessive, they may cause the molecule to change, leading to chemical reactions that that allow the energy to dissipate to some extent.

Well, that all depnds on which hat I am wearing. As a radio engineer I know that the noise power generated in a matched load on the input to an amplifier is KTB (K is Boltzman's constant, T is the temperature in Kelvin and B is the bandwidth in Hz). So I can determine the temperature of the load from measuring the noise power.

If I put another hat on, I can point an infra-red detector at a person and convert the level of IR radiation detected to their skin temperature.

If I am a quantum physicist then I am going to link entropy to temperature. Entropy is a measure of the disorder in a system, an increase in entropy represents and increase in disorder. In a liquid there is a degree of disorder in that molecules move around. As the liquid freezes and forms a solid the molecules are locked into an ordered pattern. If the liquid becomes a gas then the molecules move around more and the system becomes more disordered. So we can define temperature in terms of entropy. In The Restaurant At The End Of The Universe everyone sat around "watching entropy run down"'

The definition of temperature relates to the zeroth law of thermodynamics; so called because the other laws had been numbered before scientists realised they had been using this more fundamental law implicitly all along. Simply stated the law says that if two bodies are in thermal equilibrium with a third body then they are in thermal equilibrium with each other. Thermal equilbrium means that there is no net flow of heat and therefore the bodies are at the same tempertature.