When non-admin users try to edit their own templates using the toggle template bar they are unable to do so as the "Edit" button/link is not available to them.
The users are failing the write operation table level ACL on sys_template table. Specifically this out-of-the-box ACL with sys_id of "8f3271b2c0a8016501fab9eb78d21e95".
1) Navigate to System Security > Access Control (ACL)
2) Open the record with sys_id "8f3271b2c0a8016501fab9eb78d21e95"
3) Make sure that it has the OOB values. If not, change them to these values:
Admin overrides: checked
Name: Template [sys_template]. None
Requires role: template_editor
Condition: User is (dynamic) Me
Script: make sure this is blank
4) Save the change
If the above ACL is OOB then users should have no problem editing the templates. The ACL might have not been updated to OOB due to either having been customized before and did not receive the OOB update when upgraded or the OOB update was skipped entirely during an upgrade.