Here’s my definition: the internal energy of the stuff in a box is a useful quantity, and one can call it E. But E is the energy needed to assemble the stuff in the box if you start with an empty box of the appropriate volume. This makes physical sense, and it’s perfectly fine for calculating things related to, say, anything that happens in a vacuum. Or anything that happens in a rigid box.
But we live in a very large atmosphere, we mostly do experiments at constant pressure. If you take a flexible baggy and assemble its contents, you need the energy to make the contents (that’s E) and also some extra energy to displace air to make room for the contents, and the latter part requires extra energy equal to P (the constant atmospheric pressure) times V (the volume of the bag).
So we give E + PV the fancy name “enthalpy”, and it turns out to be useful. Maybe there’s a Legendre transform somewhere, but I’ve never seen any use for it other than to try, poorly, to convince someone of its existence. And it’s genuinely awkward — energy and enthalpy are fairly general physical quantities that one could, in principle, measure, and one already needs to start constraining the system to think of them as functions of anything sensible.
And then the HVAC industry seems to have borrowed the term “enthalpy” to mean, roughly, “temperature and humidity”. And “energy” means “temperature” or maybe “heat” but probably actually means “enthalpy, but only the thermal part and not the chemical part”. You’ll be lucky to find any math at all, let alone a Legendre transformation. Don’t get me started on “pressure”.