A user with ability to edit their own posts but not others couldn’t edit if they weren’t listed as the first author of a post. Typically their role would be that of Author. I believe by default Authors have the publish_posts capability. I was using the user-role-editor to add customize roles. The publish_posts capability was not granted to Authors and thus a problem.
If Authors have the publish_posts capability by default this problem won’t appear. It only becomes a problem if they don’t have the publish_post capability. (The publish_post is for granting/denying permission to publish new posts/pages.) As mentioned, the system was checking the publish_posts capability to allow or deny editing of an already published post. The problem is that this is the wrong capability; the edit_published_posts capability is for allowing/denying permission to edit already published posts.
