The singularity in a non-rotating, non-charged black hole is as you say. It’s like in a finite amount of time you “run out of time”, like there isn’t any more time on that trajectory.

The singularity in a rotating black hole is entirely different but the interior of classical Kerr (rotating) black holes is one of the most controversial if inconsequential topics in theoretical physics because there are reasons to believe (without real proof mind you) the Kerr solution is unstable inside the inner event horizon so that whatever happens in there is not what that theory says.

And of course black holes are quantum objects which might actually have an “interior” entirely different from the classical picture.