I am running a Gmail add-on and now would like to publish this Add-on from the AppsScript editor.
Why does the add-on-type only show "Google Docs,..." but no Gmail?
This is Google's official documentation on how to publish an app https://developers.google.com/gmail/add-ons/how-tos/publish#publish_an_add-on. If it's a private app to be used within your domain you can publish it yourself otherwise you have to submit it for review.