My confusion is whether heat refers to a process or the energy in
It refers to energy in transit, not to a process. Multiple processes can be involved with heat transfer. For example, for gases you can have heat transfer by means of a constant pressure process or a constant volume process. Similarly, work refers to energy in transit, not a process.
More importantly, and missing from your two definitions, is that heat is the transfer of energy due solely to temperature difference. The statement "temperature difference" is key to the definition of heat and is missing from both of your definitions. It is what differentiates heat from work. The latter does not require a temperature difference.
I thought the temperature difference was just a fact abt heat that
comes from the 2nd law of thermodynamics and not something defining
Temperature difference does not define heat. It is a minimum necessary (but not sufficient) condition for energy transfer to occur in the form of heat.
You can have a temperature difference between objects and no energy transfer in the form of heat if there is no mechanism that allows the transfer to occur. Conduction requires contact between the objects, convection requires the movement of a fluid in contact with the surface of an object, and radiation requires an unobstructed path between objects. If the two objects at different temperatures are isolated from one another, there can be no energy transfer in the form of heat.
More importantly, it uses "temperature" to define heat and temperature
itself is defined as being "some property" that becomes equal when
there is no net flow of heat. Wouldn't that make it circular?
It doesn't use temperature to define heat. It states that temperature difference is a condition required for energy transfer in the form of heat to occur. The temperatures do not become equal when there is no net flow of heat. As already indicated, a difference in temperature does not necessarily result in heat flow. However, there can be no heat flow if the temperatures are equal. In my opinion there is no circular reasoning here.
Hope this helps.