Fix WebSocket handshake error handling and unnecessary token refresh

[agners]

Proposed change

Fix two issues with WebSocket error handling when Core's WebSocket
endpoint isn't ready (e.g. when the http integration fails to load):

1. Unhandled exception: connect_with_auth/connect raised
   HomeAssistantAPIError for handshake failures, but fire-and-forget
   callers only catch HomeAssistantWSError (a subclass, not a parent).
   This caused "Task exception was never retrieved" errors. Fixed by
   raising HomeAssistantWSConnectionError instead, which is the
   correct exception for connection-phase failures.

2. Unnecessary token refresh spam: The retry loop in
   connect_websocket caught the broad HomeAssistantAPIError, so
   connection errors (not just auth failures) triggered token
   invalidation and refresh. This caused "Updated Home Assistant API
   token" to be logged on every failed WebSocket message. Fixed by
   narrowing the catch to HomeAssistantAuthError.

Also raised the log level for failed fire-and-forget WebSocket commands
from debug to warning for better visibility.

Type of change

• Dependency upgrade
• Bugfix (non-breaking change which fixes an issue)
• New feature (which adds functionality to the supervisor)
• Breaking change (fix/feature causing existing functionality to break)
• Code quality improvements to existing code or addition of tests

Additional information

• This PR fixes or closes issue: fixes #
• This PR is related to issue:
• Link to documentation pull request:
• Link to cli pull request:
• Link to client library pull request:

Checklist

• The code change is tested and works locally.
• Local tests pass. Your PR cannot be merged unless tests pass
• There is no commented out code in this PR.
• I have followed the development checklist
• The code has been formatted using Ruff (ruff format supervisor tests)
• Tests have been added to verify that the new code works.

If API endpoints or add-on configuration are added/changed:

• Documentation added/updated for developers.home-assistant.io
• CLI updated (if necessary)
• Client library updated (if necessary)