What does the Owners Checkbox control?
The Owners checkbox used to indicate that members are "workers" on a project instead of just "participants", and would show up in "Owners" drop-down menus on Workitems and other assets. The Owners checkbox became mostly obsolete with the adoption of member lookups instead of Owner dropdown menu. There are few UI pages such as Member Tracking page and Member Capacity page that still honor it though, and so the Owners checkbox should be used until member lookups are implemented app-wide.
* The Owners Checkbox cannot be accessed or set via the API. It must be manually checked via the UI.