LSP server initialization failure reported as timeout

### Describe the bug

In some cases, when an LSP server fails to initialize, the logs report the error as a timeout, instead of the real process launch failure. For example this session failed to launch the server due to a configuration error, but is reported as the following:
```
2026-04-22T20:45:35.358Z [ERROR] LSP csharp server for /testbed: failed to start, cleaning up: Error: LSP csharp server for /testbed:: Server initialize timed out after 60000ms. The language server may need more time for initial project analysis. Server exited with code 1.
```

This leads a copilot to believe the server timed out, and missed the real process initialization failure:

The debug logs do capture the actual error, but that is not surfaced.
```
2026-04-22T20:44:35.521Z [DEBUG] LSP csharp server for /testbed: [stderr] Could not execute because the specified command or file was not found.
```

Additionally, the `lsp test ` command doesn't seem to really test the server - it seems to report success if the process can launch (for example run using a trampoline like dnx), even if it fails seconds later due to an error. Ideally the /lsp test would actually verify that the `initialize` / `initialized` request / responses are successful.

### Affected version

v1.0.51-1

### Steps to reproduce the behavior

Create an LSP server that fails seconds after launching the process.

### Expected behavior

Copilot receives the actual server launch error, not a timeout unless the server actually times out.

`lsp test ` actually attempts verifies an LSP server can be initialized.

### Additional context

_No response_

Start at the `lsp test ` entry point and the LSP server initialization and timeout handling described in the issue. Reproduce a server that exits after launch, compare the debug stderr with the reported error, and verify that initialization succeeds or that the actual launch failure is surfaced instead of a timeout.

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