Now, I created a custom object and then created a custom tab for it, in that I selected default ON only for system admin profile, after that I did not include the tab in any of the applications. Now when I go to the sales application or any application for that matter and click on the more drop down option next to all the tabs on the right, the custom tab is not in the list but if I select personlize you nav bar option and go to "All tabs" and search for the tab I had created, I can see it in the list and add it to the nav bar from there.
The tab is directly visible to the Administrator in every application as it is set to default on and no need to explicitly include it in any of the applications.
The tab will only be visible to the Administrator in "All Tabs" unless it is included to any of the applications.
A tab cannot be set to default on if it is NOT included in any of the applications.
I feel the correct option should be the second one according to my logic.
Would love someone else's insight on this as well.