Can I rename a Microsoft Teams free site's subdomain.onmicrosoft.com? - azure-active-directory

Question

I have a Microsoft Teams free account that I created under an earlier organization name that I now wish to change. This is because my second Teams site undesirably reveals that it exists under the original organization name of my first Teams site.
Now, inviting users to my second Teams site inadvertently discloses to them that I also run the first Teams site. It comes up during certain Microsoft authentication screens. I don't want them to see that; it's distracting. Although both are non-profits, one organization has nothing to do with the other. A new user entering my second Teams site by invitation may feel confused when, upon initial entry, they're presented with my first organization's name.
I've since learned that I can indeed change my original organization name. Creating my first MS Teams site implicitly created the organization by that name within a personal Azure account that uses my credentials. It's at https://portal.azure.com, and my first Teams site shows up in there. It appears as a group within Azure Active Directory (AAD). My personal directory itself bears that initial name of my organization. The same name was automatically applied to the group representing my first Teams site.
Now, the directory itself is identified both by
the organization name, and
a corresponding subdomain name.
Both were the same, except the subdomain had no spaces embedded in it, obviously.
While I can change the directory's organization name, I don't see how to change the URL subdomain name (e.g., MySite.onmicrosoft.com) for my Teams site.
I know Microsoft Teams users aren't ever exposed to that technical subdomain information in normal everyday use anyway. However, that original name does become revealed when a new user is invited to the new Teams site. On the Android app, for example, upon initial login and setup, the new Teams user is asked to tap on the organization name. And there was my first disappointment, because it was the name of the other unrelated organization! After tapping it, users are then led to the correct Teams site that I did intend to make available to them.
That's what prompted me to want to change my organization name.
I was successful with that by changing the organization name under Properties for my directory. THis resolved the issue for the newly authenticated Android Microsoft Teams user.
However, I cannot see how to rename the subdomain itself. And that's important, because PC users at least (or those trying to enter my Teams site from a web browser) are in fact presented with a permissions prompt where that undesired subdomain name appears!
Is there a way to rename the subdomain?
POSTSCRIPT - Guess what? I can create new directories in AAD alongside my original one! What if I move my Teams group from the unrelated directory to a new one I create? Would that be safe? Will my Teams site still be functional?

Answer 1

To answer your edit. a teams organization is associated with a specific azure active directory, even if you have a second directory, i don't believe there is any way to "move" things across. you would have to create everything from scratch.
to answer your general question, you would probably have to create a new directory as you described, then create a new teams, replicate the information, remove all the users from the old teams. and remove them from the original directory. then invite them to the newly created one.
The reason for this is, if you invite someone, you are essentially adding their email / login to your azure tenant. of they have been invited to multiple teams tenants. then when they log into teams there is functionality for them to switch between all the directories that they are a guest of. so they will be able to see it. the only way to remove that is to delete the guest users from azure ad.