The XCOM promptly crushed part is canon, though I think they were implied to have been sold out rather than just brutalized on mission. The ADVENT = EXALT part requires one or both sides to have different methods or goals than originally seen, since EXALT wanted the alien's tech and humanity's ascension but without the aliens pulling the strings.
I agree that EXALT != ADVENT, partially because it was explicitly stated and also because ADVENT is more "global police force" than covert Illuminati.
I think EXALT's goals in EW were incredibly vague, though, and even their relationship with the aliens. They evidently tried to steal the alien's psionic human experiments, but they waste so much resources on antagonizing and sabotaging XCOM. And they seem to have incredible amounts of MELD to huff.
I guess they're intentionally mysterious, but I always assumed that they had an uneasy alliance with the aliens. With the psionic convoy debacle being an instance where they got greedy, or failed to cover their tracks.
The more feasible notcon is probably that EXALT has always existed, but they never officially organized into an anti-alien, anti-XCOM force because the conflict was over too quickly to make it feasible. So they're around, but not relevant to anything visible.
This is how I see it too, that they always existed but they never mobilized. They copied a lot from XCOM, like the geosphere and probably laser weaponry (even if they get lasers first somehow, I suspect they stole the concept).
They could still be active in XCOM 2, infiltrating the resistance. Assisting, for now, while covertly gathering supporters and resources. Heck, they could even have operatives in ADVENT, preparing for a coup from that direction - but needing the aliens removed first.
I was always disappointed that Enemy Within, while a lot of fun, really didn't fulfill its name. I wanted to deal with traitors and paranoia, and not like in the base defense. (Though I love its cutscene a lot)
Since this is an alternate universe, perhaps the aliens crushed XCOM in their first encounter, and the next few as well. Its perfectly possible to have the world panic and lose after like 5 missions, and EXALT only showed up for me right when i was about to start serious laser gun production. Perhaps EXALT realized that they didn't really need to do much, and then lay low and greeted the aliens. After that, a quick trip to a Public Relations expert tells them they need to change their name, it sounds too cult-like. Hence the formation of ADVENT, and EXALT (Now ADVENT) gets the world power it always wanted in exchange for being sneaky, race-betraying buggers. ADVENT is EXALT, its just the former is the patters end goal.
Sure... except it was explicitly jossed as-stated by whats-his-face the lead designer (Solomon?), check early-ish in the thread.
I haven't checked the exact quote but I bet it means EXALT didn't mobilize, not that the organization retroactively stopped existing. Likely EXALT was just a project of the organization, like XCOM is a government project.