Heat is very definitely the kinetic energy of molecules. No matter how complex and detailed the definition, no matter what particular terminology you use, there is no other way to look at it. (Somebody is bound to think of a context to illustrate that I've overstated the case, but ignore them and just accept that for any situation you'll ever encounter in your life on this earth, heat is just the kinetic energy of molecules.)
I'm not sure about the definition of internal energy. It probably has a very definite context-dependent definition that I'm not going to bother looking up, but I don't think it is useful in a simple context. Even if we got the definition right today, it is certainly at a minimum vulnerable to being misremembered at a later date. For instance, say internal energy
*is* just kinetic energy. Next month am I going to remember that internal energy doesn't include the energy of the chemical bonds of the molecule? Will I recall that it doesn't include the atomic energy? I don't trust my memory that much, so I prefer to stay with the familiar everyday concept of kinetic energy. If it were wrong to say heat is the kinetic energy of molecules then I'd make an effort to get it right, but heat
*is* the kinetic energy of molecules.
Percy, this is not how heat is defined in physics. While you can have your own definition of heat if you want (it's a free world), it's usually considered a good idea to stick to old definitions to avoid confusion. What you are describing as heat is actually closer to the definition of internal energy which includes not only the kinetic energy of molecules but also any potential energy due to the interaction of these molecules (not too important for gases, but essential for understanding liquids and solids)
Sylas had it right. Heat is defined as energy in transit in either of three possible ways.
1) radiation (that is by definition EM energy and yes, that may include energy in the visible spectrum)
2) conduction (that is energy being transfered directly by colisions between molecules when two bodies come in contact)
3) convection (that is energy being caried by the flow of a fluid - no convection happens in a solid)
cheers
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