Dogmatic doesn't mean the same thing. The most general term might be something like "magical thinking"—we use cargo cult not to refer to people holding a belief unquestioningly, but to refer to people going through the motions of a ritual without having a strong rational basis for believing it will have the desired effect, and even more specifically it implies that the ritual was borrowed from someone who actually did understand the science behind it and knew how to apply it correctly.

Even "magical thinking" doesn't capture all that nuance.

> we use cargo cult not to refer to people holding a belief unquestioningly, but to refer to people going through the motions of a ritual without having a strong rational basis for believing it will have the desired effect, and even more specifically it implies that the ritual was borrowed from someone who actually did understand the science behind it and knew how to apply it correctly.

I don't really see why this distinction is worth preserving. Dogma (i.e. arbitrary and not-rationally-based belief) seems to be the objectionable part here. Where the dogma originates seems less important.

If you don't see why the distinction is worth preserving, then you are free to simply stop using the term yourself.

No, the objectional part about cargo culting is not belief, it's the practice of the ritual. It would be objectionable even if no dogmatic beliefs were associated with it.