My understanding of it is that massive objects do not affect the speed of light at all, not even black holes.

They bend space itself meaning that light will always travel at the same speed, even in proximity of a black hole, but over a distance that becomes, from the local point of view, more and more vast, to the point where, seen from the "undeformed" space outside, it looks as if it took it longer to traverse that stretch of space (because we cannot perceive and have a hard time even imagining the fact that "there is more space" there), or it doesn't come out of it at all.

Maybe someone more into these aspects can tell me if I am oversimplifying or outright wrong.

Short answer, yes but not really.

Long answer, nothing affects the speed of light. Light always travels at the speed of light, and at no other speed. However, the path light travels is affected by any massive body. From the perspective of a distant observer, a massive body may seem to slow light down, but that's not what's happening. Instead, the massive object is literally bending space in such a way that the path that light takes - the shortest distance between two points (actually the shortest time between two points) - seems longer and hence the light seems to take longer to travel between those two points.

With a black hole, space is bent so extremely that all straight paths between any two points below the event horizon never point out of the event horizon. In other words, light inside a black hole is constantly traveling at the speed of light, but space itself wraps back around on itself. So the light has nowhere to go but inside black hole. In fact, all paths beneath the event horizon lead only to the singularity at its center. For the purposes of this, I'm ignoring rotating black holes, which work almost the same way.

A massive body that isn't as dense as a black hole will similarly bend space so that light travels what seems like a longer and slower path when viewed from a distance. Unlike a black hole, however, the straight paths ultimately point away from the massive body.

Another way to think of this is to imagine that space itself is literally flowing into a black hole, and at the event horizon, that space is flowing towards the center of the black hole at light speed. So anything traveling in that space, no matter how fast it's moving, is being pulled along with that space toward the center of the black hole. It's like a fish swimming against a very rapid current - at a certain point, no matter how hard the fish swims upriver, it's going to end up downriver as it's being carried by the current.

Light always travels at the same speed. So, nothing slows it down, not even a black hole.

What mass does is warp space-time. This appears to "bend" light (the light is not actually bending but rather space-time is) that travels past it. Think of a stretched piece of plastic that has a metal ball placed in it. The plastic will dip in towards that ball. A marble rolled in a straight line on the plastic will, when getting near the metal ball, strart curving with the depression of the plastic. So the marble (analogy for a light photon, or could be a planet orbiting a star) is actually travelling in a straight line but the plastic is warped under it.

At the event horizon of a black hole, space-time is warped by the exact amount to prevent light from escaping.

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No, the speed of light is a constant. Time however, is affected by gravity

Just a note to all those saying 'nothing affects the speed of light' - Yes, light always travels at c in a vacuum. Light traveling in a denser medium like water or glass is considerably slowed down.