IMO, an axe has to have several features to be considered a 'hawk.
1.) A straight haft. Early axes had a straight haft, so this in and of itself isn't enough.
2.) Friction fit with the head put on from the bottom of the haft, so that centrifugal force will make the head tighter, not looser. Again, early axes also had this construction.
3.) It has to be light enough and have the proper balance to fight with. The weight, plus #1 and #2, makes it as good a weapon as a tool.
For instance, my GB Hunter's axe has a very thin, hawklike head, very light, penetrates very well, even has a good beard for hooking. But, the haft is just not quite right for using as a weapon, although you could make a good go of it.
I'd probably not be quite so strict as to weight to length ratio or demands of a short cutting edge as Vec -- as I'd consider one with a 3-4" cutting edge a hawk, provided the balance was right, although Vec would probably argue that it's more of a battle axe at that point -- but his 1oz per inch of length is a damn good rule of thumb.