[RPG] What exactly is considered “light activity” for the purpose of the “Aspect of the Moon” Eldritch Invocation

It's first worth mentioning that the Sage Advice you cite is now obsolete; errata have been released which contain a correction so that long rest does now require sleep as a general rule:

A long rest is a period of extended downtime, at least 8 hours long, during which a character sleeps for at least 6 hours and performs no more than 2 hours of light activity, such as reading, talking, eating, or standing watch.

The Basic Rules (which still give the original version) and the PHB (which is updated in newer printings and on D&D Beyond — payment required) don't say exactly what happens if you continue to not sleep, other than not getting the benefit of a long rest, but Xanathar's Guide adds rules for this: DC 10 constitution saving throw to avoid a level of exhaustion after 24 hours of being awake, and the DC goes up by 5 for each additional 24 hours.

For some reason, there's no level of exhaustion between "so tired you can't move" and "completely dead". It seems like at some point "you fall asleep" would be on the chart, but, nope. Death it is.

With that in mind, a Con 10 character without proficiency in that save has a 16.5% chance of being fine after 48 hours, but only a 0.82% chance of feeling normal after 72. (Of course, a character who made it through the first 48 has a 5% chance of one more night where everything seems fine.) After that, though, the DC is impossible, so it's just a matter of counting down to sudden, permanent sleep.

You'd think a level 20 barbarian with 20 Con would be much better at ignoring the pesky limitations of mortal flesh — and indeed the first night doesn't even require coffee, and the second presents just a 15% risk of getting a level of exhaustion. But as the DCs go up to impossible levels quickly, outside the bounds of 5E's math, everything goes downhill suddenly. Even with a streak of good rolls (early on, where they even matter), the eleventh night is certain death for even the most hearty hero.

Apparently sleep is significantly more important in D&D than in the real world; in 1965 a high school student stayed up for 11 days, with no apparent long-term harm. And there is no indication that this student was particularly proficient in Con saves, or even had class levels (beyond those available in high school). Scientists have monitored subjects kept awake for 8-10 days, also with no harm that couldn't be fixed by "one or two nights of recovery sleep" — which is much faster than 5E's slow recovery of one exhaustion level per night.

I find this somewhat disappointing: I know that D&D is not a good physics simulator, but I expected it to be pretty accurate on the subject of all-nighters.

In seriousness, I think the "and now you're dead!" thing is primarily there to make players take this seriously and not just never sleep except when spells need renewed. Personally, it's never come up in a game that I've run, but if it did, I would add "hourly con saves to stay awake" at level 4 and increase that to "every minute" at level 5, with further exhaustion only happening if some effect causes it or if you are forcibly kept awake.

This should work, if it's the last thing you do before going to sleep.

Outside of combat, we typically don't account for actions, but this invocation would require keeping track of whether you do something that would be an action. For example, you can eat during a rest, but that would use an action if done in combat, so it would break the invisibility (because the magic doesn't care whether you call it an action).

However, if the warlock goes into an enclosed, dark space, lies down, invokes the magic, and then goes to sleep, yeah, they can be invisible until waking up.

A caveat about doing this during a short rest

(not actually the question, but it seems relevant)

The short rest rules mention "tending to wounds" as a possible activity, and allow regaining hit points by spending Hit Dice, but don't actually say that you have to tend to your wounds to regain hit points. I would argue that the rules imply that wound care is an expected part of resting, and if you insist on spending the entire rest invisible instead of caring for your wounds, you can't spend Hit Dice. But depending on how your table treats the rules, and on how you interpret hit points, this could go either way.