I'm most likely going to offend someone in the course of writing this entry, so consider yourself forewarned.
The ordinary definition of 'vegetarian' in common use in the US today is someone who doesn't eat products that are the result of animals being killed. I think some people who are less aware of how it's used would just think of it as not eating animal flesh, but for people who are vegetarians and those familiar with the issues, I think it's much more along the lines of the former.
Where this tends to get confusing initially is how you define animal. I would say that conventionally, this definition excludes people who eat fish (and shellfish, etc.). Certainly Wikipedia thinks it does. Such people are sometimes called pescetarian, but that's a silly word because a pescetarian, if it were analogous to 'vegetarian', would be someone who mainly eats fish. However, some people don't think fish are animals, so they are fine with the definition of vegetarian as someone who doesn't eat animal products, and consider themselves vegetarian, because fish are not animals. This is probably true in an older sense (possibly originating in Biblical use) but seems a bit funny as a claim these days. So there's not really a good word for this habit (just like there's no good word for people who don't eat red meat, except 'I don't eat red meat'). They can say "I don't eat meat, but I do eat fish", but if they also don't eat, for example, rennet-made cheese, then it just gets too complicated.
However, the definition of vegetarian also officially excludes me, because I'm lazy and sometimes eat food containing gelatin, and I eat cheese made with rennet, occasionally. And the stomachs used for rennet come from veal production. I stopped eating veal when I was quite young because I found out how it was made before I found out about CAFOs in general. So eating cheese made with rennet is not only not technically correct, it is not at all ethically correct within my own standards. The same is true of the cheese and eggs I occasionally eat, namely that they're not ethical within my own system, which involves (among other things) avoiding CAFOs. I just haven't managed to get my food behavior up to my own standards.
So I really don't feel like I can claim that people who eat fish are somehow not vegetarian, yet I am. Especially because most of them are conscientious eaters of fish, and eat things like wild-caught tilapia which are just fine ethically (in my ethical system) and for the planet. If I weren't lazy I might have better grounds for this assertion, but still, it would be a somewhat strident claim to make. Even if they only eat fish for convenience, I certainly understand that it can be difficult to get over that hump, just as I am having difficulty getting over the cheese & eggs hump. I didn't give up fish totally until well after I had given up all the other animal-killing things that I currently don't eat, and I hardly even like fish.
There are also many commonsense reasons why it's easier for those people to talk about themselves as being vegetarian. The category is a convenient one when expressing their dietary restrictions to other groups or institutions, because they can easily not eat fish in any given situation (and often they usually don't eat fish), but they can't easily eat other meats, so they are not omnivores. And they usually have a similar perspective on ethical eating choices and a similar social experience of having a marginalized diet, so in many ways they are part of the exact same group. So while I fundamentally think that such people are not vegetarian by common use of the word, neither am I, yet I believe that we should all be allowed to describe ourselves that way, while introducting caveats when appropriate. However, here I will put my foot down and say that if you eat things that everyone agrees are animals, like chickens or cows or goats or many other things, then you are not vegetarian. You can be semi-vegetarian, veggie-friendly, or flexitarian if you want, but you are not vegetarian.
Pretty much the same thing applies to veganism, except that the definition excludes all animal products. But the same issues come up. Are you going to worry about bone-char sugar? Isinglass? Are bees animals? (We don't employ any other insects to make our food, so this is a bit of a one-case issue.) If you eat honey should you call yourself only vegetarian and suffer the confusing caveat-full consequences of that? I would say that the same applies here for people who are less strict as vegans as for people who eat fish: they're largely inside the category vegan, so it makes more sense to stick with that. However, I will put my foot down here, as I did above, and say that anyone who eats dairy products and eggs is vegetarian, not vegan. No reason to get confused on that major point when a more applicable category is available. Even though I eat vegan food most of the time and drink soymilk and not milk, use Earth Balance and not butter, I'm not vegan by any stretch of the definition.
It should, as an aside, be noted that my ethical system is actually much more along the lines of ethical omnivorism than vegetarianism. Many vegetarians object to animals being killed for food at all. I object to animals suffering needlessly, and I object to a lot of other things that are involved in the way food is industrially produced, which includes the way some vegetables, like corn and soy and potatoes, are grown. I don't mind animals (or plants) being dead so that I can eat them, because that's just how nature works. What I'd really like to do is eat local, organic, humanely-raised food of all kinds, mostly plants (after Michael Pollan's "Eat food. Not too much. Mostly plants."). But that's pretty hard. I'm working on all aspects of it, but for right now I think it's easier to try to give up as much in the way of CAFO-produced animal products as I can manage, and call myself vegetarian because it's the best-fit category.
Ultimately, I'm certainly not going to go around being the vegetarian definition police. Not only is it annoying and unjustified, but also I seem to remember something about a beam in my own eye...