When it comes to things like gravity and the electromagnetic force, masses aren't reduced-but with nuclei the mass difference is noticeable. What about nuclear forces makes them capable of putting mass into binding energy?
The only difference is that the strong force is stronger; meaning that binding energy can be a notable fraction of the total energy. You write
which is not quite true, but then let on that you know by continuing
which is the whole answer. Masses are reduced by electromagnetic or gravitational binding, but not by enough to be noticed.