How are hot and cold defined?
The concept of temperature arises from the zeroth law of thermodynamics:

If two systems are in thermodynamic equilibrium with a third system, the two original systems are in thermal equilibrium with each other. Basically, if system A is in thermal equilibrium with system C and system B is also in thermal equilibrium with system C, system A and system B are in thermal equilibrium with each other.

From here, many hypothetical scales for temperatures are stablished.

It's said than one body has higher temperature than another when there is a heat exchange from the former to the latter.

But hot and cold are just sensations or feelings defined by people. Something is called hot when it's at higher temperature than our body (~37 °C), so a positive heat exchange will take place on contact. The opposite for cold (heat transference from our body to the cold object).

So, you are right, the higher the temperature of an object, the hotter it will be, but the concept hot is just to define the thermal perception.