Here is what I have found so far about the term "Ainur":
(Root) AYA-N, treat with awe/reverence. Cf. especially ainur, “the Holy Ones” ...This ainur only appears in plural since after the Creation all those known were máyar. Máyar includes Valar and their lesser kin, but not those who either did not take part in the Great Theme, or else did not enter into Eä. So The Ainur is used only of the Spirits before Creation, or of those unnamed who are not concerned with it. ...
(Root) MAY- Original sense probably “make” (in artistic sense ..). The noun máya, maia was applied to all the angelic spirits created by Eru before the making of Eä. ...
(Root) MAY- [Cf. máya/maia, name of the Kin of the Valar, but especially of those of lesser power than the 9 great rulers. ...]
These come from two of four references to a (root) MAY-. The four references give three different meanings for the root.
The case for using “Ainu” as the generic word for this kind of being is that the word is used that way in the Silmarillion.
"If ever in their dealings with Elves and Men the Ainur have endeavoured to force them when they would not be guided, seldom has this turned to good."from "Of the Beginning of Days"
"So died ... Elwë Singollo, King of Doriath, who alone of all the Children of Ilúvatar was joined with one of the Ainur."
from "Of the Ruin of Doriath"