"Birds" is a taxon (Aves is the scientific name for it). In modern biology, all taxa are defined to be "monophyletic lineages" or "clades", which means that in principle they comprise one ancestral population and all descendants. No taxon would be defined nowadays as you define it since your definition does not necessarily identify a clade. A valid monophyletic taxon might be coincident with a set of organisms identified by a definition such as yours, however.

Edit: The Wikipedia article has a section on the definition of the relevant taxa: https://en.wikipedia.org/wiki/Bird#Definition