Even if elementary, I'm afraid I am still not comfortable with the concept of heat. I can picture heat as some kind of incompressible fluid that is transferred from a body to another in such a way that its flux is opposite to the temperature's gradient. But I can't reconcile this "picture" with the abstract, axiomatic concept of heat, when it is introduced more rigorously in the context of the first law and of conservation of energy. We define heat as a form of energy transfer that is not explainable in terms of purely mechanical work. So, is heat a kind of fluid or is it just a way energy can be transferred and, more importantly, what is really the "bridge" between the two concepts? I think that historically people simply noticed that when a hotter body is placed in contact with another, the temperature of the latter tends to rise and so we say that some quantity of "heat" has passed from one to the other. This is very intuitive, but then why can't we say that a cup of tea "contains" a given quantity of heat? Any thought is appreciated.
Heat is like electricity. In either case we talk about a fluid which moves from high potential to low, and this serves to abstract a messy physical situation. Electricity can really describe a flow of electrons, or a flow of ions. Similarly, heat can really describe the propagation of molecular vibrations (conduction), or a flow of molecules (convection) or photons (radiation).
The abstraction is only useful when one of these mechanisms is happening, so heat and electricity only exist when they're being transferred from one system to another. The higher-potential system makes heat/electricity, it flows into the lower-potential system, and then it's absorbed into that system. Then it simply becomes charge/energy.
So: There's always a thing which is transferring energy from one system to another. But we don't want to get into the details of what that thing really is, so we just call it heat and discard it once the transfer ends.
The Caloric Theory of Heat was abandoned. Heat is not a noun, although it is still used that way, because of how it has been engrained in language. Heat is a verb. That is, as you stated, how it has come to be known in canonical thermodynamics. An object doesn't contain a certain amount of heat; it has a certain temperature. But an object can certainly heat another object; but that heating isn't done by passing some fluid between the objects as was thought years ago.
Check out the Caloric Theory of Heat from Wiki: Caloric Theory
as well as Ralph Baierlein's Thermodynamics Book, for a nice explanation of "heat" and the noun/verb understanding.