You question is mostly good, except , change MEANING to MEANINGS.
These are all somewhat similar, but there are some nuances. In my opinion:
1 he’s intelligent = straightforward, intelligent/smart.
2 He’s intellectual = he's probably intelligent and he likes to spend his time on 'intellectual' activities, e.g. reading serious books, philosophy,studying history, listening to jazz and classical music, etc.
3 He’s logical = he's probably intelligent, at least in ways related to reasoning, etc. He may not be the 'artistic/creative type'.
4 He’s rational = pretty much the same as logical.
5 He’s bright = general term meaning intelligent.