Since we deal with heat in thermodynamics, let us try to see what this quantity really is.
Our first and most frequent encounter with heat is through hot and cold objects that we come in contact with in our daily lives. Therefore, our intuitions about heat are guided essentially by temperature, and not (quite) in terms of energy flow into, or out of, our bodies through the contact area. In other words, what we feel is a temperature difference. [Actually, our perceptions of hot and cold also depend on the rate of energy/heat flow through the contact area; for example, a piece of wood and a piece of metal would give you a different qualitative sense of “hotness” or “coldness” even though they are at the same temperature.]
Thus, appeal to our intuitions can take us only so far. Let me try to excerpt here how textbooks handle this topic.
Peter Atkins in The Laws of Thermodynamics: A Very Short Introduction (Oxford University Press, 2010):
… In everyday language, heat is both a noun and a verb. Heat flows, and we heat. In thermodynamics, heat is not an entity or even a form of energy: heat is a mode of transfer of energy. It is not a form of energy, or a flow of any kind, or anything of any kind. Heat is the transfer of energy by virtue of a temperature difference. Heat is the name of a process, not the name of an entity. [Page 22]
Richard H. Dittman and Mark W. Zemansky in Heat and Thermodynamics (7th Edition, McGraw Hill, 1997):
Heat is either internal energy … in transit […]. During the process of heating, energy flows from one part of a system to another, or from one system to another, by virtue of only a temperature difference. When the flow has ceased, there is no longer any occasion to use the word heat or the symbol Q, because the process is completed. All that remains after heating has been completed is a different state of the system, that is a new value for the internal energy … Consequently, it is incorrect to refer to the “heat in a body,” just as it is correct to speak of the “work in a body.” The processes of working and heating are transient activities that lead to a change of the energy found in a system. All that endures is the new state of the energy. The energy of a system cannot be separated into a mechanical part and a thermal part, just as you cannot analogously identify some water in a lake as originating from this river and other water from that rain. The river and the rain have lost their meaning, but the new water level endures.
What I think is important is that when heat (energy) is transferred into (or out of) a system, it causes not only an increase (decrease) in the system’s energy, but also an increase (decrease) in another state variable called the entropy, , associated with the system. Formally, of course, we have the following form of the second law:
I will add more, as and when I find something new.