The word "cultist" wasn't common at the time. Lovecraft's productive period was 1908~1936. In the Google book corpus, the word “cultist” only pops up around the 1920s, and then again only gets popular considerably later; see here. It must have felt like a neologism back then, and we all know how Lovecraft was fond of archaisms rather than novelties. Compare with “worshipper”.
As for why people today use the word to describe his worshippers—well, now we have the word in general use, and it fits. After all, a “cultist” is just someone who belongs to a “cult”, and a “cult” is:
n. 1. (offensive, derogatory) A group of people with a religious, philosophical or cultural identity sometimes viewed as a sect, often existing on the margins of society or exploitative towards its members. (Wiktionary)
This certainly could be used to describe the sects craving the Crawling Chaos or bowing to the Blind Mad Sultan. What's more, even though HPL didn't use the newer word “cultist”, he did use the older root “cult”—38 times, even, in the collected fiction. The first time he used it was in The Hound (1922):
Alien it indeed was to all art and literature which sane and balanced readers know, but we recognised it as the thing hinted of in the forbidden Necronomicon of the mad Arab Abdul Alhazred; the ghastly soul-symbol of the corpse-eating cult of inaccessible Leng, in Central Asia.
Some other interesting passages include:
The Rats in the Walls (1923):
Sir William, standing with his searchlight in the Roman ruin, translated aloud the most shocking ritual I have ever known; and told of the diet of the antediluvian cult which the priests of Cybele found and mingled with their own.
The Call of Cthulhu (1926):
The writing accompanying this oddity was, aside from a stack of press cuttings, in Professor Angell’s most recent hand; and made no pretence to literary style. What seemed to be the main document was headed “CTHULHU CULT” in characters painstakingly printed to avoid the erroneous reading of a word so unheard-of.
The Shadow Over Innsmouth (1931):
[…] and she assured me that the rumours of devil-worship were partly justified by a peculiar secret cult which had gained force there and
engulfed all the orthodox churches.
It was called, she said, “The Esoteric Order of Dagon”…
The Shadow Out of Time (1934):
Some minds recalled more than others, and the chance joining of
memories had at rare times brought hints of the forbidden past to future ages. There probably never was a time when groups or cults did not secretly cherish certain of these hints. In the Necronomicon the presence of such a cult among human beings was suggested—a cult that sometimes gave aid to minds voyaging down the aeons from the days of the Great Race. […]