There is no "equating the two observers".
> when speaking of 'formation time', we are speaking of the formation time by observers at infinity
No, we aren't. The hole isn't formed at infinity. The time that is being talked about for observers at infinity (or more correctly, very far away) is the time it takes light rays, coming outward from the object that is collapsing into a black hole, to reach those observers. The curvature of spacetime near the horizon is such that it takes longer and longer for those light rays to get out to the observer far away, the closer to the horizon they are emitted. At the horizon, outgoing light rays do not move outward at all, due to the curvature of spacetime there, so they never reach the observer far away--hence that observer never sees the horizon form. But, as I said, that is an optical illusion caused by the curvature of spacetime.
> What I understand is that all observers must agree on reality, thus on whether the BH forms at all
All observers agree on observable events and measurement results that they can all observe, yes. But if an observer cannot observe a particular event (such as the horizon forming) because spacetime curvature prevents light rays from that event from ever reaching him, then he can say nothing at all about whether that event happens or not.
> evaporation seems to complicate this classical GR view understood for a while
It complicates it in the sense I said before: that observers far away, instead of never seeing infalling objects cross the horizon, now see those objects cross the horizon (more precisely: see light rays emitted outward by those objects when they crossed the horizon) at the same time they see the hole evaporate (more precisely: see light rays from the hole's final evaporation). It does not mean what you claim, that infalling observers will see the hole evaporate before they fall in. The spacetime geometry of an evaporating black hole (i.e., including quantum effects) is different from the spacetime geometry of a classical black hole that never evaporates. The statement that a distant observer never sees an object cross the horizon is only true of the latter, not the former.
> I defiantly conclude they don't exist (or I'm missing something major) :)
The second is the correct choice. You need to stop being defiant and start learning what the models of black holes in physics actually say.
> black holes in nature would be something akin to a extremely time-dilated "Object trap" without singularties (although perhaps it must gain an apparent horizon?)
To us far away, an apparent horizon looks the same as an event horizon, and works the same--all the things I said above would still be true (at least for the time we have been observing--see below for how long it would take for differences to become observable) if the objects we observe and call black holes actually have only apparent horizons (because of hypothetical quantum gravity effects that, as I've already said, are an open topic of research), not event horizons. And all of your criticisms would still be wrong. (In at least some of the hypothesized quantum gravity models, there is no singularity and no event horizon, only an apparent horizon; but to us far away, the object looks and works the same into the very far future; it takes something like 10^70 years for the difference to be observable.)