These heat/temperature/energy issues are tricky to explain - I quite agree.
As to your
Temperature is how much heat there is in a certain amount of matter
I am not sure that is strictly accurate. Surely it depends o the nature of the matter? For example a polystyrene block the size of a housebrick will have much less heat in it than a standard housebrick when they are both at the same temperature.
So how about
:
Temperature is about how much heat there is in a particular piece of matter?
JF