Does a person have to be wealthy and powerful in order to be considered successful?

That depends on what you define 'success' and 'rich' as. Success is often considered only in terms of wealth (i.e. rich issuccess) but other successful people may be only as rich as most other people, for example athletes that win at the Olympics or inventors that develop something simple and cheap.

Successful people are always rich, but not necessarily monetarily.

A successful person can make lots of money at a given profession,
and it is said he or she is rich.

However, a person who works at a job helping people, but the
pay is low could feel rich.

He/she enjoys seeing the hope the job gives to others, for example.

The job possibly saves lives, and this line-of-work makes one feel rich.