Issue Type: Bug

If I remove a computer from the list of machines that have settings sync enabled, I cannot rename another computer to the name of the computer that was just removed. This issue also occurs on the latest insiders build.

Steps to Reproduce (The placeholder names of my-computer and my-work-computer can be substituted with any names):

1. Enable settings sync on multiple computers with VSCode
2. From one of the computers, open the panel shown by the Settings Sync: Show Synced Data command (or by clicking the "settings sync is on" message in the account menu in the sidebar) and rename the current computer to my-work-computer or some other friendly name.
3. From the same panel, rename the other computer to my-computer or a different name.
4. Switch to the other computer (the one named my-computer in this example). Make sure settings sync has pulled the lastest data and the friendly names for both computers show up in the Show Synced Data panel.
5. Delete the computer named my-work-computer (The computer that is not marked as current/the one we were last using)
6. Attempt to rename the current computer from my-computer to the same name that the computer we just deleted had (in this example, my-work-computer)
7. VSCode displays a message stating that machine names must be unique, even though a computer with the name my-work-computer no longer exists in the list of computers that have settings sync enabled.

VS Code version: Code 1.63.2 (899d46d, 2021-12-15T09:40:02.816Z)
OS version: Windows_NT x64 10.0.22000
Restricted Mode: No

Screenshot of issue:

In this case, the name of the computer I previously deleted was called Kytech-Surface. It does not show in the list of machines with settings sync enabled as it has been deleted. Attempting to rename any of the computers with settings sync to the old name produces the same issue: