The code on the Rental Library VHS says "FOXVID" something. VHS tapes in the early 80s sometimes had these codes superimposed in the upper part of the image (maybe it had something to do with the duplication process?). I know that Warner also did it, there's a similar code on the old "Nelvanamation" video that had A Cosmic Christmas on it.

This information would be overscanned out on the CRTs of the era, but watching on an LCD or a capture makes them more visible. (I think the top of the code is in that part of the image you don't see unless you "scan", i.e. rewind or fast-forward while the tape is playing.)

I seem to recall that I used to have a Magnetic Video copy of The Making of Star Wars which had a similar code that said "FOX" and some numbers.

I think the 1982 video releases must have come from the IP of the 1981 reissue that introduced the Episode IV crawl. Like most (all?) of the pre-THX pan and scan releases, it had the 1981 Fox logo instead of the classic one, but the '82 VHS I checked out from the public library about 5 years ago also replaced the 50s fanfare with that John Williams re-recording done for ESB. Was that on theatrical prints of the '81 reissue too? (All of the non-THX P&S versions I'd seen had the logo replaced, but not the fanfare.)

And there's a flaw in your argument about different IPs being made in '77 - all pre-ANH prints and bootlegs we've found have these marks, whether from 35mm or 16mm, stereo or mono, domestic or foreign, with the original or revised end credits. If I had to guess, the marks were introduced when the lab did the '77 interpositive, or internegative, or whatever was used to make the 35mm and 16mm prints pre-ANH. (Did the print shown in Baltimore have them, I wonder?)