What ARE birds?

are mammals birds?
are reptiles birds?
are insects birds?
are fish birds?
are echinoderms birds?

All mammals are birds!
Not all reptiles are birds.
Some insects are birds, although I do not believe that all are.
Some fish are birds.
I've found a single citation that suggests one or two echinoderms may be birds, but I can't find any definitive information without paying like a hundred books to buy a book. For now, let's say none are, as the vast majority are not.