Okay, how's this: All flies are lures but not all lures are flies.
Over the years, I've participated in many (a lot) of discussions/threads on this very subject and given it some thought. Aside from what is stated in individual state and locale fishing regulations, here's my opinion and it's just that. My individual opinion and you may agree, disagree or feel it needs some tweeking. 'A fly imitates a water born or water found insect. It is tyed on a single pointed hook. It may have some minor added weight wrapped under the body but the attachment of a weighted or even an unweighted bead head makes it a 'lure'. A lure can be reasonably, effectively cast and often even fished with typical spinning equipment but a fly cannot. A true fly can only be cast because it is carried by the fly line. The fly itself offers no aid to the cast.' Just some of my thoughts and many of those contained in posts above are good too.
Allan
P.S. Again, barring any specific state DEC regs, maybe the best way to determine 'fly' or 'lure' would be to paraphrase what Supreme Court Justice Potter Stewart said about 'obsenity' (again paraphrasing) 'I may not know how to define it but I'll know it when I see it'.