refactor(telemetry): track btw side question as tool_call event - #2257

Related Issue

N/A

Description

Replace the lightweight input_btw telemetry event with a structured tool_call event so the /btw side-question flow shows up alongside the rest of the tool-call observability (outcome, duration, error breakdown) instead of just a fire-and-forget input counter.

1. Structured telemetry for btw side questions

Problem: /btw previously emitted a single input_btw event at the Shell layer, which only told us that the user typed /btw … — not whether the side question actually returned an answer, how long it took, or why it failed. That made it impossible to spot regressions like LLM-not-set, model attempting tool calls instead of replying, empty responses, or transport exceptions.

What was done:

• Remove the track("input_btw") call from Shell._main_loop (it was only counting input, not outcomes).
• Wrap execute_side_question in a try/finally that emits tool_call with tool_name=btw, outcome, duration_ms, dup_type=normal, and error_type on failure.
• Cover all error paths with distinct error_type values: LLMNotSet, ToolCallDenied (LLM tried to call tools instead of answering on both turns), NoResponse, and the exception class name for uncaught failures.

2. Tests

Problem: Telemetry regressions are silent — without explicit tests, future refactors to execute_side_question could easily drop the track call or mis-label an outcome and no test would catch it.

What was done:

• Add TestExecuteSideQuestionTelemetry in tests/ui_and_conv/test_btw.py with one test per outcome branch: success, LLMNotSet, ToolCallDenied, NoResponse, and an exception path.
• Each test asserts the event name, tool_name, outcome, error_type presence/absence, and that duration_ms is recorded.

Checklist

• I have read the CONTRIBUTING document.
• I have linked the related issue, if any.
• I have added tests that prove my fix is effective or that my feature works.
• I have run make gen-changelog to update the changelog.
• I have run make gen-docs to update the user documentation.