I believe your confusion lies in the colloquial definition of heat. In thermodynamics, the heat ($Q$) refers to the energy ($E$) of the system that isn't available to do work ($W$) i.e. it has units of $\mathrm{Joules}$.

The temperature of a system is a defined quantity. It is the amount of heat the system increases by per unit of entropy ($S$), i.e:

$$ T = \frac{\mathrm{d}Q}{\mathrm{d}S}\;\;\;\;\ \langle \mathrm{Kelvins} \rangle $$

Your next question probably is what is entropy? Briefly, it's the number of ways you can have your system at a particular energy. This is a strictly positive number.

Qualitatively and in conclusion, the temperature scales how much energy can be transferred via the system in a given ensemble of states.

I believe your confusion lies in the colloquial definition of heat. In thermodynamics, the heat ($Q$) refers to the energy ($E$) of the system that isn't available to do work ($W$) i.e. it has units of $\mathrm{Joules}$.

The temperature of a system is a defined quantity. It is the amount of heat the system increases by per unit of entropy ($S$), i.e:

$$ T = \frac{\mathrm{d}Q}{\mathrm{d}S}\;\;\;\;\ \langle \mathrm{Kelvins} \rangle $$

Your next question probably is what is entropy? Briefly, it's the number of ways you can have your system at a particular energy. This is a strictly positive number.

Qualitatively, the temperature scales how much energy can be transferred via the system in a given ensemble of states.

EDIT, oops better actually answer the question:

The temperature and heat of a system are related, but can evolve independently and non-linearly with energy thanks to the joys of entropy.