CiviCase Users Unable to Edit & Add Activities on Cases They've Been Assigned
To me this reads like a bug - either the description is misleading, or the permission isn't functioning. You should log this at issues.civicrm.org, including all the detail you've added here (and a link back to this page).

Your options from there are basically:
1) Wait for the core team to fix it.
2) Pay for the core team to fix it.
3) Recruit a freelancer or an agency to fix it for you.

Message me if you want me to look at this in more depth for you.