If heat is energy being transferred between hot and cold water, why would it be incorrect to say that temperature measures heat?

For the same reason that the amount of water you pour into a container doesn't tell me the depth. The depth in a teapot and a swimming pool are different even if you pour the same amount of water into both of them.

The temperature change of a boxcar load of lead and a gallon jug of Oxygen are different, even if you transfer the same amount of heat into both of them.

it is the most commonly accepted one