You are splitting hairs, and you know it. IBM honcho Jack Sams asked Gates to buy the rights to QDOS from the guys in Seattle. No, Gates did not buy it "from" IBM, but he wouldn't have bought it at all if IBM hadn't asked him to - and if IBM itself didn't want to dirty its hands by buying it themselves (yes, they could have outbid Gates at that time, hard to believe now, I'm sure). That is the famous $50k payment, made right before the release of the IBM pc so that IBM didn't have to worry - haha - about any problems from some upstart software guys. My history is not "garbled," it is correct in its essence, if you want we can all go to an encyclopedia together and quote it at length.
My understanding is that Allen is the guy who knew about QDOS in Seattle, and that the rest was a bluff by Gates to IBM.
Gates did not OWN QDOS when he sold it to IBM as MS-DOS. At the time, IBM thought they were dealing with a much larger company than Microsoft was. Gates had written a version of BASIC for the Altair, Apple, Commodore and several other companies, so had a track record, but he had never written an Operating System. IBM approached him as a consultant to discuss what would be appropriate for their soon to be released PC.
He suggested they talk to Gary Kildall about acquiring CP/M (Control Program for Microcomputers). They did try but Kildall was apparently not available (out flying his plane, it's said) and his wife and lawyer refused to sign a non-disclosure agreement.
Frustrated with Kildall, IBM offered Gates the contract to WRITE an OS. Bill Gates had a contract in hand and had to deliver something he did not own. He had no Operating System. With that contract in hand, Gates licensed Q-DOS (Quick and Dirty Operating System) from Tim Paterson of Seattle Computer Products.
Basically, Q-DOS was a reverse engineered, simplified version of CP/M. Gates and company then refined and improved on Q-DOS and delivered PC-DOS 1.0. IBM found that, after testing, product contained over 300 bugs in about 6000 lines of code... it also appeared to contain unmodified code from CP/M (vehemently denied by Paterson even as late as 2004) which was a legal problem. IBM demanded a re-write. Gates bought the next version of Q-DOS, 86-DOS outright (Exclusive righs) from Paterson's company and Microsoft hired Tim Paterson to do the re-write. PC/MS-DOS was born.
Was this wrong? Nope. Just smart business.