Not sure if this is definitive, but it gets into a decent amount of detail!
https://github.com/trunkmaster/nextspace/blob/master/Documentation/OpenStep%20Confusion.md
I personally always preferred NeXTstep and NeXTSTEP. If I had to chose between them NeXTSTEP was my favorite for some reason.
As for OPENSTEP, for me, it was always the all caps version. OpenStep, seemed almost 3rd party for some reason to me.
What's your favorite variant?
Oddly enough, that guide has some errors!
- "NextStep" was the de facto name at first and even appears on Steve's slides during the NeXT Introduction.
- IBM actually did port NS 1.0 to AIX (https://www.osnews.com/story/29649/the-curious-case-of-nextstep-on-aix/), but Steve wanted more money for 2.0 and they (sensibly) pulled out. This was when IBM and Microsoft were still teammates on the OS/2 project and barely made sense business-wise, but had it lasted longer it might have been a very interesting rival to CDE/Motif.
- The "NeXTSTEP" stylization did indeed die off during the transition to NeXT Software, but the versions listed are wrong; it was around from 2.0 to 3.1PR1. 3.1 final (the first x86-capable version) was all-caps.
- No one called it MachOS! It's just "OPENSTEP for Mach".
- "OPENSTEP for Windows" is actually called "OPENSTEP Enterprise," or OSE. There were also HPUX and Solaris versions of OSE, the latter of which is not to be confused with Sun's OpenStep for Solaris 1.0/1.1.
Detailed list of version and product names here, of course: https://docs.google.com/spreadsheets/d/1bjpLWLxp6j891Nu8u6NzvwnpN5_NuUAJei283JVrW7U/edit?gid=0#gid=0