Refactor agent-service

### Task Summary

Umbrella for refactoring the `agent-service` codebase into clean, single-topic changes — no runtime behavior change. The work is split into small, independently reviewable PRs that build and pass tests on their own, landed in dependency order. Each sub-task is tracked as a sub-issue below.

**Sub-tasks**
- #5749 — consolidate the WebSocket message-type framework → #5751
- #5942 — remove the dead checkout/headChange path → #5930
- #5748 — centralize backend DTOs, metadata, endpoint config, and auth → #5928
- #5750 — redesign the sync-execution result and error model → #5927

Supersedes the earlier all-in-one refactor (#5564) and the broad cleanup (#5563); agent-service test-coverage (#5266) is folded into each sub-task.

### Task Type

- [x] Refactor / Cleanup
- [ ] DevOps / Deployment / CI
- [x] Testing / QA
- [ ] Documentation
- [ ] Performance
- [ ] Other

Start by reading sub-issues #5749, #5942, #5748, and #5750 in their stated dependency order; this issue names no files or test paths. The work is done when each independently reviewable refactor builds and passes tests without changing runtime behavior, with test coverage folded into the sub-tasks.

Written by the indexing model from the issue text.