To answer your question, we can take a look at the Administrator user guide, you can find the definition of owner:
Owner — The person(s) responsible for managing the specification in a particular status. As owner of the workflow, you:
■ Receive items in your Action Items list. You are responsible for assuring that the assigned parties perform all necessary actions or reviews.
■ Can advance the specification in the workflow process by clicking Workflow in the upper right corner of the specification page.
so if user in the group doesn't have edit right defined for that stage, and also not selected exclusively in the WFA edit permission, then owner wouldn't be granted with edit rights. however, user with [SPEC_ADMIN] role will be granted with read and edit rights no matter what.
Hope this helps.
To expand a little on Jessie's comment. We do have an "Owner" token in the permissions grid. This allows you to not have to select the same user multiple times, once in the owners grid and once in the permissions grid. So if you want to grant the owner edit rights you just need to make sure that token is added for the statuses you are interested in.