Hi Faith,
This is a very interesting subject.
First let me correct a misconception. Birds do not appear earlier in the fossil record than dinosaurs. They do, however, overlap dinosaurs by quite a lot, which is a point some creationists use as "proof" that birds cannot have evolved from dinosaurs, though they look over the fact that this would not say they didn't evolve at all!
I am sure you can see the fallacy of such an argumnet... it's just another version of the old "If humans evolved from apes, why are there still apes?" argument.
Our knowlwdge about the evolution of birds from therapod dinosaurs is still very young... and we're learning more on an almost monthly basis.
The deabte about the exact evolutionary origin of birds is still fairly heated. Many argue that the Theropods are the ancestors of modern birds and that the earliest birds show clear evidence of such a relationship.... for example, scientists now know archaeoptyrex did not have an opposed rear toe, so it's feet looked exactly like a dinosaur's foot and not like a modern bird's foot.
http://news.nationalgeographic.com/...201_archaeopteryx.html
In fact, although archaeoptyrex has usually been considered a dinosaur-like bird, some now consider it a bird-like dinosaur.
Of course, we know about feathered dinosaurs, but the issue does get murkier.
The earliest "birds" seem to appear earlier in the fossil record than the best examples of unambigusously feathered dinosaurs, though cladisitc research indicates that many of the earlier fossil theropods we do have were probably feathered, but we just don't have the skin impressions to prove it. A small minority think the poorly preserved fossil of Longisquama insignis is evidence that birds and dinosaurs had a common ancestor, rather than birds descending from dinosaurs themselves. Those that are proponents of this idea hold that "feathered dinosaurs" are actually early flightless birds, though this idea has it's own complications.
Of course, this gets to your main question, which is one of classification. It's important to realize that classification is a tool invented by humans, and not imposed by nature. We find classifcation useful as a way of describing relatedness, but nature is not at all concerned about defining precise boundaries to assist us. And it only gets more complicated as we find more fossils which fill in the gaps of the record. So, is archaeoptyrex a bird-like dinosaur, or a dinosaur-like bird? It certainly has some features we simply don't usually associate with modern birds. But it is certainly more bird-like than what we tend to think of as a dinosaur.
So the short answer, I think, is that scientists are finding it difficult to draw a bright line between dinosaurs and birds. If you polled 10 paleontologists about this subject, you probably find 8 different answers and 2 who say it doesn't matter.
I hope I helped.