tl;dr: the only allusions that are clearly referring to PRECISELY what we're talking about, all seem to indicate the 17th century version. The ellis text sounds different from the bespectacled,virtually anthropomorphic bird-mask.
Ah see I can't give you that. Actually based on my feverish reading of plague related wiki articles I believe that the bird mask as we know it now is 17th. One of the articles had a image of an engraving depicting an earlier plague doctor and the beak mask looked to just be a beak over the nose and mouth not a full face mask.
Also any idiot could have been a plague doctor the only requirements seem to be thinking you knew what you were doing and taking peoples money for doing it.
Fourteenth century plague doctors who wore a bird-like mask were referred to as "beak doctors". where are you getting your information there is the damn direct quote and citations again.
Ok so first off the engraving I thought was from earlier was actually the 1721 engraving from Marseilles. Still a beak mask, still doesn't look like a bird though.
And now a game I like to call dueling wikis
"Some plague doctors wore a special costume, although graphic sources show that plague doctors wore a variety of garments. The garments were invented by Charles de L'Orme in 1619; they were first used in Paris, but later spread to be used throughout Europe The protective suit consisted of a heavy fabric overcoat that was waxed, a mask of glassed eye openings and a cone shaped like a beak to hold scented substances."
"In the seventeenth and eighteenth centuries, some doctors wore a beak-like mask which was filled with aromatic items. The masks were designed to protect them from putrid air, which (according to the miasmatic theory of disease) was seen as the cause of infection."
Both from the wikipedia on plague doctors.
"The costume is also associated with a commedia dell'arte character called Il Medico della Peste (the Plague Doctor). The character's mask is associated with the early-17th century French doctor Charles de Lorme."
From the wiki on plague doctor costume.