Yes, this is correct. If you look at the Schwarzschild solution to the Einstein field equations you'll find that there are two apparent singularities: one at t

Yes, this is correct. If you look at the Schwarzschild solution to the Einstein field equations you'll find that there are two apparent singularities: one at the center of the black hole, and another at the event horizon. However, the apparent singularity at the event horizon is not a true singularity because you can perform a coordinate transformation in which it disappears. Mathematically, this is doing what you describe: going from the reference frame of an external observer to one who is falling into the black hole. To the outside observer you appear to take an infinite amount of time to fall in, but from the perspective of the person falling in, it happens in a finite amount of time. The other singularity at r = 0 is different, though. It is a true singularity because there is no coordinate transformation you can make in which it disappears.