DOS/360 was announced in 1964 and first delivered in 1966--my copy of the S/360 Assembler manual dates from 1966 and mentions DOS, TOS and BPS. I don't recall if it mentions BOS, however.
Early S/360 software implementation could be kind of rocky. For example, S/360 COBOL didn't initially include any direct-access disk support--the compiler understood the syntax, but there was no run-time support. I have a publication that describes a bunch of assembly-language routines to provide the missing DASD support. "ENTER LINKAGE" phrase...
So did IBM coin the term "DOS"? I think so--they tended to use different language than the seven dwarfs. So terms like "VTOC" and "OS" were pretty much IBM-ese, as I think, were tems like "linkage editor".