My reading of
https://www.copyright.gov/title17/92chap1.html#117
I would say that when downloading from an illegal download site, you aren't copying the program you purchased. In many cases the downloaded copy isn't identical to the original, it may have been patched to work round drm etc & that causes it's own legal issues. Philosophically you could argue it's still the same game, but the law is clearly written with the intention of you making a copy from a legally purchased one. Even if the end result is bit identical, there is no exemption that explicitly says you can obtain a copy elsewhere. So it seems rather unsafe to rely on an implicit exemption.
Copyright and licensing are different, some copyright holders will give you permission to make as many copies as you like but then license you to use a set number of those copies at any one time. If you have that agreement then you don't need to rely on fair use.
Your DVD example isn't equivalent as that is a clear copyright breach even if you purchased the original as combined works need permission from the copyright holder.
Cases where producing a receipt would be a defense just don't go to court, so a judge would never get to rule on it. It's especially unlikely as a judge would likely rule against awarding any damages, because there was no lost sale.
The DRM one is a separate issue from where I sit, though I have seen rulings go either way as things might be necessary and even that seems to include provisions for making something work.
Bit identical copies is a potential issue, and that is before we consider whether my having say an emulated copy of whatever N64 game that contains at least theoretically (sometimes we get say a 1.3 that never hit
https://www.zeldaspeedruns.com/oot/generalknowledge/version-differences but we will assume no new censorship or bugfix issues cropped up) identical game to the straight N64 version. That said it is an issue and presumably why many places say roll your own as that little bit of safeguarding. Don't particularly see it as a dealbreaker though as much as an issue the would be downloader has to be concerned with not unlike might get a virus instead.
I can also see a path to an argument of said differences if they are more simple bugfixes and censorship, and not something like a gold edition with all the expansion packs/DLC or indeed a translation*, being some kind of allowable or seriously entertained (even more so if there was a return program, recall or something at the time, maybe even a simple acknowledgement of error which the v1.1 might be a tacit version of -- such a thing presumably not being free to issue or create) or in the case of the BIOS if it conferred no particular advantage (maybe if the difference, functional or more academic, itself between versions did not rise to a copyrightable act being a further contemplation). That however would be an argument you get to make that might fail to persuasive.
*though if the baseline is already established then NTSC English <-> more or less identical PAL English might be where I go next. If it was just a find and replace color with colour type and drop to 50fps for PAL deal then philosophically speaking I don't see any issue. Those parallel translations where Europe might well have got the far superior version then being a different matter. Line could get very hazy as well as things are removed to comply with local standards (think gambling machines removed in pokemon).
Combined works as you call it, derived works as many other places would know it generally, and adaptations as the link to the law text you provided seems to want to call it might well be a separate issue.
If I am playing sneaky/clever lawyer though it seems more a matter of timings
"or authorize the making of another copy or adaptation"
Assuming I am the one doing the censor work with the DVD editor (or computer program if sticking with the narrower restriction there).
Your call/payment to me initially renders you an owner of the baseline DVD (or program) from my stack of them and thus I am now your authorised agent in its modification. You might not be able to sell the modification onwards at a later date (whoo nice bonus for my business I guess) without permission of the holder but that would seem to be the obvious workaround.
Might also go for downloading a cracked version (assuming said download does not have a nice crack separately as many will) if the you then consider said chosen download site your authorised agent. They might be in trouble at some level but that tends to be less your problem and more theirs (that fairly recent supreme court case about printer toner probably being a good example, though patents more than copyright was the thing used here).
"If you have that agreement then you don't need to rely on fair use."
If doing the 1 cart/dvd = 1 game (or presumably BIOS if continuing with the earlier topic) then that would change things, and I have not seen a game provide me with so many chairs/CPU cores/concurrent users/... like so much OS or productivity software.
Equally if fair use is less restrictive than what the holder might care for -- same as any other contract that might go against what the law says you can include in a contract.
Implicit vs explicit law and restrictions thereof... getting into the fun legal philosophies today (others playing along there is a constant debate in legal philosophy, which also varies between countries and sometimes parts of them, about whether laws merely say what you can't do and all else is presumably legal, and prescribe what it is you can do with anything falling outside that being illegal). UK law tends to be instances of restricted actions, which is why we get idiocy like the response to legal highs laws wherein they decided they could not keep up with "twiddle one useless non functional group and give that a spin", US generally does that too and indeed going against it would be rather counter to their whole history.
I would not disagree with a statement along the lines of "a cautious legal type might not do well to say yeah all good" to such a scenario as discussed here like they might "you are driving the speed limit while in total control under perfect conditions, nothing wrong with that". Equally that law is probably ancient and written by people more ancient and less understanding of computers (or ghost written by industry) and thus did not anticipate such a scenario or actively left it somewhat murky.
If I am sticking with harms done, unfair competition and whatever else that determines things in these sorts of cases I am still lacking anything to say "you must grow enough tech and tech skills and do it yourself/have your computer guy do it for you".
Amusingly enough if the download site had some kind of "by downloading this you agree you are an owner of a copy of the software" type deal that might be further fun.