refactor: merge internal/sync into internal/garmin, regroup api files and routes

Garmin auth/sync routes move under /api/garmin/*; sync.Service becomes
garmin.Sync with garmin.SyncConfig/ClientConfig; applog becomes
internal/log; the test mock moves into the garmin package as MockClient
(breaking the test-only import cycle the merge created); stale test
URLs and type names updated to match.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>

docs/superpowers/specs/2026-07-26-improve-synchronization-design.md

@@ -22,7 +22,7 @@ only production caller of the core logic) calls `backfillCore`/`incrementalSyncC
 Only `internal/sync/service_test.go` still calls `Backfill`/`IncrementalSync`, which is the only
 reason they're not already flagged as unused by the compiler.
 
-Since this plan already restructures `service.go`'s progress model, remove these two dead
+Since this plan already restructures `sync.go`'s progress model, remove these two dead
 exported methods (and their now-inaccurate doc comments) as an early task, rewriting the tests
 that called them to exercise the same behavior through `FullSync` or the `*Core` functions
 directly (same package, so unexported functions are still directly testable) -- before any of
@@ -120,7 +120,7 @@ able to report it; the frontend type was just never updated to match).
 ## Frontend design
 
 `GarminConnection.tsx` keeps its connect/MFA/Reset-all logic and buttons untouched, but loses
-its `syncStatus` polling, `syncProgressLabel` helper, and all inline sync-progress/last-sync
+its `garminSyncStatus` polling, `syncProgressLabel` helper, and all inline sync-progress/last-sync
 JSX.
 
 A new `SyncModal.tsx`: