This is a difficult question to answer. If I were to allow using discretion on anything within reason, there would be some here who would take that too far and we would get even more invalid results than we currently get on here. And that sort of thing has really been out of control lately, so I'd hate to do anything to encourage that even more.
My own approach, and one I think is probably a reasonable one to take, is this: if the cause of the problem is something that could have just as easily happened in a real competition as happened to you, then you should treat it as strictly as would be treated in a competition. So things like starting a OH solve in a mini-Guildford with both hands should probably be a DNF; it's true that it's not an official event, but if it were, the WCA would certainly consider that a DNF. A similar case I've had is with 3BLD where I forgot to start the timer before looking at the cube. I've had 2 DNFs in official WCA competitions like that; it only makes sense that should also be a DNF at home. (I can't believe how often I make that mistake!) And as a good example, clock DNFs are very common and should still count as DNFs at home, whether it's due to being off by one or due to solving to the wrong 12 o'clock (both of which I've done multiple times in online competition, and both of which I've done in official competitions).
On the other hand, if it's something due to weird conditions of solving at home, I think more discretion should be reasonable. If I have an FMC solve which was correct on paper, but I copied it into the computer incorrectly, I think it's perfectly reasonable to correct it and count it as a valid solve. If it was not written down correctly to begin with, that's different though - that should be a DNF. For your example of not starting the timer because you were on the wrong tab, that feels like something you should have another chance at, and I have done this. I think that should be treated as an extra; as I have said a couple of times before on here (probably many years ago), I think you should feel free to generate yourself an extra scramble if needed (using a decent scramble generator) for the sorts of things that would generate extras in a competition. I also think extras are perfectly reasonable for cases where the circumstances are something that are due to competing online and that could not have happened in a real competition.
I hope that seems reasonable, and I hope it helps.