Black holes - is any of these points of view more accepted? I'm a physics enthusiast and I'm quite interested mainly in astrophysics.
Recently, I've come across this very interesting video from PBS Spacetime, where they state that:

A black hole is the collection of events that -for outside observers- don't happen at all, even waiting for an infinite amount of time.
The event horizon of the black hole is a region in space-time that contains the collection of the last events visible from a fallen object world-line.
A black hole doesn't suck stuff in, and it isn't black because not even light can escape it, it's just that the space-time geometry is so curved that there is no simply radially out way

This seems like a really good relativistic explanation and so far I think I grasp the important part of it.
The problem now, is that * virtually every other didactic source ever* states the known definition:

A black hole is an object whose mass is so dense that it collapses on itself, forming an infinite dense spot, whose gravitational pull is so strong nothing can escape it, not even light

Okay, this is the traditional definition I'm used to, but this leaves me wondering: why do we say that not even light can escape it.
If the space-time geometrical impossibility of a radially out way is true, it should be implied that nothing (including light or whatever particle) can get out.
I feel like I'm missing something, or that many sources are either wrong or oversimplified.
So the question is: is any of those points of view more valid than the other? why?
Disclaimer: Not a physicist
 A: Both of these are fine heuristic definitions of a black hole, and they are correct enough for most applications. They are both insufficient, however, when your goal is to formulate mathematical proofs (e.g. Penrose-Hawking Singularity Theorems) involving black holes. The mathematical definition of a black hole is much more subtle and clarifies many of the ambiguities present in statements like 'not even light can escape from a black hole'. Escape to where? 

If the space-time geometrical impossibility of a radially out way is true, it should be implied that nothing (including light or whatever particle) can get out.

What is it about the geometry which makes it impossible to escape? It is certainly possible to conceive of trajectories which begin inside the event horizon and exit the black hole. The problem with these trajectories is that they are neither time-like (meaning that a massive particle could travel along them) nor light-like (meaning that a massless particle, such as a photon, could travel along them). In order to preserve causality, nothing can travel along a space-like trajectory. So, there are certainly radial ways out of the black hole, but causality demands that not even light can travel along them.
A: The traditional statement is wrong. For instance if you have a shell of collapsing matter then the event horizon that stops everything, even light, from being able to leave forms when the shell still encloses a ball of radius $2GM.$ Which is before any infinite density happens.
And you can also have infinite density without a black hole. Make that shell infinitely thin but still have a total mass of $M$ and give the shell a radius greater than $2GM.$
A: 
I'm a physics enthusiast and I'm quite interested mainly in astrophysics. Recently, I've come across this very interesting video from PBS Spacetime, where they state that:
A black hole is the collection of events that -for outside observers- don't happen at all, even waiting for an infinite amount of time.

There is some truth to this. The outside observer never sees the infalling observer cross the horizon. His "finite proper time" takes forever to happen.

The event horizon of the black hole is a region in space-time that contains the collection of the last events visible from a fallen object world-line.

Given the above, this description is reasonable. Check out the mathspages formation and growth of black holes and pay special attention to the frozen star.

A black hole doesn't suck stuff in, and it isn't black because not even light can escape it, it's just that the space-time geometry is so curved that there is no simply radially out way.

It does suck stuff in, and it is black because light can't get out. The "no radial way out" is a fairy tale. See this question.

This seems like a really good relativistic explanation and so far I think I grasp the important part of it.

It isn't. It's a really bad relativistic explanation. For a good relativistic explanation you need to read what this guy said.


The problem now, is that * virtually every other didactic source ever** states the known definition: "A black hole is an object whose mass is so dense that it collapses on itself, forming an infinite dense spot, whose gravitational pull is so strong nothing can escape it, not even light"

That's not right either. Gravity is only there because the "coordinate" speed of light varies. This goes to zero at the black-hole event horizon. And it can't go lower than that.

Okay, this is the traditional definition I'm used to, but this leaves me wondering: why do we say that not even light can escape it.

Because of the obvious answer : because light can't escape a black hole. That's why it's black. This goes all the way back to John Michell in 1783.

If the space-time geometrical impossibility of a radially out way is true, it should be implied that nothing (including light or whatever particle) can get out.

Light can't get out, but it's nothing to do with the spacetime geometrical impossibility of a radial way out. It's because the speed of light at the event horizon is zero.

I feel like I'm missing something, or that many sources are either wrong or oversimplified.

You are, and they are.

So the question is: is any of those points of view more valid than the other? why?

There's some issues with some of the things you've quoted. Go back to the original Einstein and IMHO you'll understand black holes no problem.
