D7 errors given when authenticated user changes password but pwd still changes
Something doesn't add up with what the information you have given. When a user edits their own profile, the default is to simply reload the same page.

You said that you have added a link on their profile (view) page, that points to the user edit page. 

I'm guessing that the following is part of the link you've added:

destination=admin/people

If there is, then you need to remove this. Anytime there is a destination=[SOMEPATH] in the URL and a form is submitted, it will redirect the user to the path given in the URL.