Well yes... And no.
"Maiar" isn't a taxonomic classification; it just signifies Ainur who went into the world but not including the 15 most powerful. So any claim that a particular individual is a Maia must of necessity imply that this individual existed before the world and took part in the Music.
That Melkor did corrupt many of the Maiar is beyond question, but on the other hand Tolkien all but shouts from the rooftops that there were other spirits in Ea, presumably created during the Music by both Eru and the Ainur, and there's a very clear reference to these in the Silmarillion:
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Therefore he gathered to himself spirits out of the halls of Eä that he had perverted to his service, and he deemed himself strong. And seeing now his time he drew near again to Arda, and looked down upon it, and the beauty of the Earth in its Spring filled him the more with hate.
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Make particular note of the phrase "looked down" here, because that's evidence that Ungoliant is one of those spirits:
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The Eldar knew not whence she came; but some have said that in ages long before she descended from the darkness that lies about Arda, when Melkor first looked down in envy upon the Kingdom of Manwë...
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That everything not otherwise classified must be a Maia is something that, I believe, comes from David Day, so that's probably a good indication of how seriously to treat it.