[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_

No files or tests are named. Reproduce the reload sequence in Chrome with bash as the default shell, then trace how the web terminal restores or selects its shell after the page reload; done means reopening with CTRL+` launches bash rather than sh.

Written by the indexing model from the issue text.