GitHub Enterprise sign-in loops with mixed personal and GHE accounts

Does this occur consistently? consistently inconsistent
Repro steps:
the sign GitHub Authentication extention stil fails when someone has both a personal and a GHE identity. Using profiles it completely ignores them and keeps trying to direct the user to standerd github for auth instead of providing a mechanism to attach to ghe. simple to reporduce , don't use your corporate microsoft machine. use a standard non domain joined machine, then create a retail github a ccount and then try and sign in with a GHE account and enjoy the death loop of log in prompts. for the real expereince add a tight timeline and pressure from PM to complete a task andn then you will feel what many are dealing with because your auth flow is broken for real people, or at the very least completely unintuitive

Action: @azure.toolInvoke
Error type: Error
Error Message: Cancelled

Version: 1.0.209
OS: darwin
OS Arch: arm64
OS Release: 25.6.0
Product: Visual Studio Code - Insiders
Product Version: 1.123.0-insider
Language: en
UTC time: Tue, 02 Jun 2026 13:13:35 GMT

Call Stack

```
u extension.js:13:53207
async yv.login extension.js:13:53711
async dc.createSession extension.js:13:64826
async file:/extensionHostProcess.js:532:21676
```

Start by reproducing the mixed personal and GitHub Enterprise account flow in VS Code Insiders on a non-domain-joined macOS machine. Trace the login and createSession calls in extension.js and the extensionHostProcess.js stack, then verify that the flow offers a GitHub Enterprise path and no longer loops or ends with a cancelled session.

Written by the indexing model from the issue text.