feat(lsp): pass `client_id` to `on_initialized`

[mrcjkb]

Follow-up to #583

[github-actions[bot]]

Review Checklist

Does this PR follow the Contribution Guidelines? Following is a partial checklist:

Proper conventional commit scoping:

• For example, fix(lsp): some lsp-related bugfix

• [ ] Pull request title has the appropriate conventional commit prefix.

If applicable:

• [ ] Tested
  • [ ] Tests have been added.
  • [ ] Tested manually (Steps to reproduce in PR description).
• [ ] Updated documentation.