"[E]nergy can be transferred by two mode i.e. Work and heat."
That's the classification we use in Thermodynamics, when considering how a system can gain or lose internal energy, U. Thus
gain in *U* = Net heat into system – Net work done by system.
U is a function of state. The evidence is as follows. If we insulate the system (to stop heat entering or leaving) and take the system from one state, $S_1$, to another $S_2$, we find we always have to do the same amount of work, whether we do it mechanically, electrically or whatever.
Now what about heat? We find that even if we do no work on a system we can change its internal energy by putting it in contact with something at a different temperature from its own. If we believe in the conservation of energy (and I do, fervently) then there must be energy entering (or leaving) the system. This – that is energy in transit by virtue of a temperature difference – we call heat.
Neither heat nor work are functions of the body's state; they are energies in transit, and we can vary the proportions of heat and work by which we effect a given change in U.
Maybe I haven't paid enough attention to your distinction between energy and a means of transporting energy. I suspect that it's not a very useful one. For example, how would you test by experiment which interpretation is correct, or does either interpretation lead to a contradiction? On the other hand we do like to be clear about our concepts… A couple of further thoughts: (1) heat as a mode of transport of energy certainly doesn't seem much like a bus transporting its passengers; it's more like the passengers themselves. I mean that passengers are people in transit by vehicle, just as heat is energy in transit by conduction, convection or radiation. (2) In the UK many Physics teachers have been persuaded that one shouldn't use the term heat at all, but should instead talk of energy transferred by heating. ['Energy transferred by heating' is rather a mouthful; can't we think of a word for it? (Joke)]