Saurischians are dinosaurs, dinosaurs are saurischians. Birds are saurischians. Birds are dinosaurs.
Since the publication of The Origin of Species, and the flurry of evolutionary classification that took off after it, birds have been recognized as some kind of Archosaur or another. Birds have always been known to have been dinosaurs.
You are certainly correct that the identification of birds with dinosaurs was proposed quite early, but it is going a bit too far to say that was generally "known" to be the case. The "Are Bird's Dinosaurs?" debate, was absolutely a topic of argument for a century and a half, but is really two overlapping debates. The first is the actual descent of birds from fossilized (proposed) ancestors, and the second is a fundamental one for evolutionary classification, how should you group species (i.e. what is a taxon)?
Linnaean taxonomy was based on grouping species by common characteristics, but this created lots of paraphyletic and polyphyletic taxons. A paraphyletic taxon includes an ancestor, but only some of its descendants as member; a polyphyletic taxon excludes the common ancestor of the members and usually some of the other descendants as well (convergent evolution is a common reason for this).
Cladistics, the use of statistical grouping methods (aided by computers), clarified things considerably starting in the 1960s, and then genetic analysis put it into hyperdrive, proving the fundamental correctness of the cladistic approach, and giving it an unambiguous standard of validity. Modern groupings invariably strive to be monophyletic.
Cladisitics actually helped clarify the interpretation of the fossil record for birds so in the 1980s biology came to recognize that a consistent method of classification, combined with a better interpretation of fossil evidence required birds to be classified as a type of dinosaur.
It really is more than a matter of changing labels. This did create a change in understanding about how to think about dinosaurs based on the surviving branch of the taxon, and about the origin and nature of birds.