The meaning did not shift as described in the OP quote. The Old French (which is the source of English) use of fysike for healing potions, and, accordingly, fisiciien for healer, dates back to 12th century, and predates its later use for the study of nature from 1580-s. More narrow modern use as in physics (that replaced natural philosophy) is even older, c. 1715, and the term physicist was only introduced by Whewell in 1836 with explicit reference to already established use for healers:"As we cannot use physician for a cultivator of physics, I have called him a physicist."
Of course, the original Greek use of "physics", as in the title of Aristotle's treatise, goes back to ancient times, but it did not filter through to medieval Europe until translations of Aristotle became widely available in 16-th century, and then the shift happened the other way. By the way, Greek physis has always had an ambiguous meaning, with nature as in cause of things being prominent, and this idea of natural causes was actively emphasized in corpus Hippocraticum:"It is not sufficient to learn simply that cheese is a bad food, as it gives a pain to one who eats a surfeit of it; we must know what the pain is, the reasons for it [dia ti], and which constituent of man is harmfully affected". This may explain the medieval European use prior to the re-introduction of Aristotle.
Nonetheless, Greeks themselves used a different word (iatros) for healers, including in Aristotle's reference to Hippocrates as a "great physician" (as it is typically translated) in Politics VII, 1326a, see Baker, Aristotle on the Nature and Politics of Medicine:
"By way of preface, I here note that the ancient Greek word iatros (ἰατρός), though variously translated as “healer”, “doctor”, or “physician”, is not simply ambiguous. Instead, there seems to have been, even in the time of Aristotle, a close semantic connection in iatros between “healer” and “medical doctor” since of course a medical doctor is someone skilled at healing. Something similar may be said of the corresponding verb iatreuō (ἰατρεύω; “to heal”, “to treat medically”) as well as the phrase technē iatrikē (τεχνὴ ἰατρική; “the medical art”, “the art of healing”). So while I will consistently translate iatros as “doctor” and (technē) iatrikē as the “medical art,” let us bear in mind that the former would have been heard as something like “expert healer,” and the latter as something like the “the expertise of healing”."