I would describe myself as vegetarian but there is a wide variety of ways to be strict about it so it's almost a useless way to describe oneself. Personally, I avoid cheese because of rennet, wine because of eisenglass, I won't eat anything with gelatin, i avoid eggs unless they come from my friends who have chickens (because I know their chickens are well cared for). I end up being close to vegan but don't really feel like that label fits me because I'm sure I eat butter without realizing it, or other milk products which can end up in places you don't expect (milk is in tootsie rolls, for example).
On the other hand I know vegetarians who just avoid meat and are fine with chicken or beef stock or gelatin.