[Bug]: Terminal shows incorrect shell type after reloading webpage

### Is there an existing issue for this?

- [X] I have searched the existing issues

### OS/Web Information

- Web Browser: Chrome version 110.0.5481.178 (Official Build) (64-bit)
- Local OS: Windows 11 Pro
- Remote OS: Debian 11 in LXC
- Remote Architecture: x86 (AMD)
- `code-server --version`: 4.10.0

### Steps to Reproduce

1. Set default shell to bash.
2. Open terminal, should be bash.
3. Click trashcan and kill the terminal.
4. Refresh page, or close and re-open tab before reopening terminal.
5. Hit CTRL+` to open terminal, will open sh terminal instead of bash -- hitting trashcan and reopening without the refresh it will open bash terminal as expected.

### Expected

Expect that default terminal is launched regardless of state when the application was last used. If bash is set to default, it should launch a new bash terminal.

### Actual

The wrong terminal is launched and the default setting is ignored.

### Logs

_No response_

### Screenshot/Video

_No response_

### Does this issue happen in VS Code or GitHub Codespaces?

- [X] I cannot reproduce this in VS Code.
- [X] I cannot reproduce this in GitHub Codespaces.

### Are you accessing code-server over HTTPS?

- [X] I am using HTTPS.

### Notes

_No response_