This seems more of a linguistic/English question than a quantum question, but I'll answer it all the same.
All four papers you've linked refer to the same concept, that of a unitary operator controlled by multiple qubits.
The first and third papers use the phrase "multiply-controlled", the second paper uses the term "multiple-control", and the fourth paper uses the term "multiple-controlled". The first two versions are grammatically sound; the third is not.
"Multiply-controlled" is correct because "controlled" is an adjective, correctly modified by an adverb, "multiply". "Multiple-control" is correct because "control" is a noun, correctly modified by an adjective, "multiple". "Multiple-controlled" is incorrect because "controlled" is once again an adjective, so it should not be modified by another adjective, like "multiple".
Personally I would use "multiply-controlled", since in context "controlled" serves as the adjective modifying eg. "Toffoli gate", as in "multiply-controlled Toffoli gate". But note that the term "Toffoli gate" itself consists of a noun ("Toffoli") modifying the noun "gate", so I really don't have any issue with "multiple-control Toffoli gate". Both are, I feel, perfectly clear about what they are referring to, and it does no harm to have both phrases in the literature.
But - sorry to contradict you - "multiple-controlled" does sound wrong to me! Even so, it's still perfectly clear what the phrase means, so my discomfort with it is really just a manifestation of linguistic puritanism. ;)
One More Note
Please note that the adverb "multiply", pronounced "mul-ti-plea", is entirely distinct from the verb "to multiply", pronounced "mul-ti-ply"! The adverb adds no semantic meaning beyond the adjective "multiple"; it's just a change in tense.