I actually sold personal computers back in the mid-80's and dealt with this stuff on a daily basis - here's the short version on all this:
The 8-bit Apple II ran Apple DOS, which was developed to make use of the "incredibly cool" floppy drive (Disk II) which Steve Wozniak had created. The Apple III (and the improved 16-bit Apple II) ran Apple DOS's replacement, ProDOS. Then along came the Lisa in about '83, which used a GUI (graphical user interface) "inspired by" the work at Xerox PARC. Xerox also developed and sold a box using their GUI (the Alto, as in "Palo Alto"), but they didn't really see it as a core part of their business. Apple learned a lot from the Lisa effort and the result (January, 1984) was the first, "insanely great", 128K Macintosh. I actually owned one of these (autographs inside the case and all) and eventually donated it to the Computer Museum.
MS DOS was based on QDOS (written by Tim Paterson of Seattle Computer Products), while MS Windows was one of at least two PC GUIs (the other that I remember was Digital Research's GEM) developed in the mid-80's to compete with the Mac. If you think Windows is a pain now, you should have seen v1.0. I was very glad I had a Mac at the time.
If you're at all interested in the early days of the "revolution", I recommend Fire in the Valley by Paul Freiberger and Michael Swaine. It was first published in 1984 (which is the copy I have) and then revised in the late 90's according to Amazon. A great read with lots of insights into the "players" like Jobs, Gates, Wozniak, and many others.
Correct. One of the others was IBM's "Top View", the other was Quarterdeck's "DesqView". I had DesqView and liked it--wow, multitasking without using TSRs!
The other was Amiga OS, which was probably the best of the lot. It did things in 1985 that were not done on the other systems until 1998. GEM wound up on Atari's ST "Amiga killer" (it didn't, it took a crooked investment banker named Gould to do that). There were other GUIs that would even work (sort of) on old systems like Commodore 64/128 and the Apple II series. GEOS by Berkeley Softworks (I think) was really quite innovative on those very limited memory computers.
Meanwhile, Commodore's Amiga could run Mac software in emulation FASTER than the same software ran on native Macs with the same processor! The Amiga could even run PC's MS-DOS software in emulation, although not quite as fast as on a PC with an equivalent processor.
The Amiga broke new ground by using dedicated sub-processors for many routine tasks that both the Mac and PCs used the CPU to accomplish. It was only later that PCs adopted this technique (first with processor/graphic cards) to increase performance. The Amiga also pioneered the library and re-entrant process approach as part of its pre- emptive multitasking system.