> but since "science" (c. 1834AD) is a newer term than "φυσιολόγο" (c. 350BC)
The word "science" is a lot older than 1834. Have a look at page 225 [0] of the 1622 edition of Richard Hooker's Of the Lawes of Ecclesiastical Politie, where you will read (my emphasis):
> The reason, why no man can attayne beleefe by the bare contemplation of Heaven and Earth, is, for that they neyther are sufficient to give us as much as the least sparke of Light concerning the very principall Mysteries of our Faith; and whatsoever we may learne by them, the same we can onely attaine to know, according to the manner of naturall Sciences, which meere discourse of Wit and Reason findeth out...
Hooker is here using "natural science" ("naturall Sciences") to mean essentially the same thing as the phrase means today. And, although that's from the 1622 edition, that section of the book was first published in 1597, and I believe that passage is the same in the original 1597 edition (Hooker died in 1600).
Earlier in the same book (also first published 1597), he says (page 193 [1], my emphasis):
> It is with teachers of Mathematicall Sciences usuall, for us in this present question necessary, to lay downe first certaine reasonable demands, which in most particulars following are to serve as Principels whereby to worke, and therefore must be before-hand considered.
Once again, here he is using the phrase "mathematical science" ("Mathematicall Sciences") in essentially the same sense as it is used today. (The point he's making there, is he sees the axiomatic methods used in mathematics as a model for theology to emulate.)
And I doubt those are the first uses of the word "science" in a way clearly compatible with contemporary English usage. They were just the earliest I could find after a brief search.
[0] https://babel.hathitrust.org/cgi/pt?id=gri.ark:/13960/t0ns23...
[1] https://babel.hathitrust.org/cgi/pt?id=gri.ark:/13960/t0ns23...