Respectfully, that's exactly what it says. Why else would someone get promoted "until they reach a level at which they are no longer competent" if one's competence increases with a promotion? e.g. Some people are intuitively better at managing people than they are at programming.
> employees are promoted based on their success in previous jobs until they reach a level at which they are no longer competent, as skills in one job do not necessarily translate to another.
That does not translate to or imply that:
> The peter principle assumes that competence in one role automatically translates to less competence in a higher role.
It is saying you will keep getting promoted until you are incompetent. You can be more competent at a higher role without violating the Peter Principle. That would be another point at which you would be promoted working your way towards incompetence. The Peter Principle just says you were competent at each step in the ladder until you got to the point you weren't promoted anymore because you are incompetent at it.
You're then take another logical leap in saying that promotion to the next level means you are competent at every job on every career path that could get you to that level. The Peter Principle only talks about your own career path.
Edit: You seem to be referring to this quote of yours:
> Why else would someone get promoted "until they reach a level at which they are no longer competent" if one's competence increases with a promotion?
My response explained why your leaps in logic don't make sense. That line just means there's a point at which their skills won't apply. Not that every rung above results in reduced competence. You can be better and better and then worse and the Peter Principle would still be applicable.
The Peter Principle is just about the changing competencies as one rises and how one hits a wall.