No, they are different things.
Before relativity, the concept of "Newtonian black holes" was considered, with a horizon which had an escape velocity equal to the speed of light. A /powered/ spaceship could still escape.
What you told me - repeatedly - was that you can't escape from a black hole because you can't accelerate to faster than the speed of light. That is simply an incorrect justification.
A correct justification is that you would need an infinite amount of energy to keep up the thrust needed to escape. Alternatively, you would need your thrust to be pushed out with an infinite total momentum.
And of course, you can say that space-time is so warped that every direction is "down", but that is much harder to understand.
I understand now that there is no escape for the spaceship once it has passed through the event horizon - as I noted, I am not an expert in this subject, and I have learned a little more during this thread. But I don't think your "escape velocity" explanation is any better than my one just because it happens to match the correct end result.
As a stand-alone reason, this is not enough. From the viewpoint of the spaceship, the inside of the black hole is not that special - it sees no discontinuity or jump as it passes through the event horizon.