First off, Yes, dinosaurs are birds. Unlike the word "fish", which was in used long before terms or concepts like "monophyletic" were invented, "dinosaur" is a scientific term that arrived around the time scientists were developping cladistic classification, and the scientists have made the choice of defining it as a clade (a theoritical last common ancestor+ all of its descendants). Therefore, any descendant of a dinosaur is a dinosaur.

For older words, the scientist definition doesn't need to be taken into account in general use, for example, the scientific definition of "berry" is famously different from it's popular definition, but you don't use the scientific definition in everyday life. But for "dinosaur", a word coined by scientists referring to something that is only known through science, it makes less sense to ignore the scientific definition.

As for dinosaurs not being birds, that is true for most, but if birds are dinosaurs, then there were some dinosaurs that were birds. There's actually two conflicting definitions of birds: If it's a theropod that could fly or is descended from one that could, and is closer to any modern bird than to deinonychus, then birds (=Aves) appeared either in mid Jurassic or in and were already quite diverse before the K-T extinction, including the enanthiornithes and hesperornithes groups, that disappeared during the extinction.

If you define it as the common ancestor of all modern birds and its descendants (=Neoornithes), then they appeared in the late creataceous.

Using either definition, it is clear that they all look more at birds than like anything else, and a layman seing one if them out of context would immediately think of them as a bird (tho maybe a strange one) rather than as a dinosaur. So unlike for berries or fishes, there would be no conflict between either the scientific definition of "bird" and the popular one. Either way, only 3 separate lineages among them survived, so the meteorite did kill whole bird species.