In iTunes Connect, how do I register an app with a Personal Team, rather than with the Organization my apple id is currently associated with?
New App with Personal Team, not Organization
Not sure I understand the question. 'Teams' are used for signing. The 'Organization' is the owner on the account.
All apps added to an iTunes Account are associated with that one account.
Are you asking about roles on that Team as part of the organization that owns the account?
Are you asking about a Personal Team associated with free provisioning?
What do you mean 'register'...? Do you mean codesign for the store or add/upload to the store?
Might want to see this document:
https://developer.apple.com/library/content/qa/qa1915/_index.html