[risu729]

Replaces #1581 and keeps its commits, plus review fixes.

Summary

• apps/extension/src/login/badge.test.ts keeps the hoisted wxt/browser mock, the root timer hooks, the describe scopes, the shared fixtures and harness. Synchronous registrar modules in apps/extension/test/login/badge/ hold the presentation, writes and navigation, persistence and page-status recording cases.
• The badge.test.ts max-lines exemption is removed from .oxlintrc.json (the entry is about 800 lines, the largest module about 400).

Review fixes on top of #1581

• Renamed the modules to *-cases.ts, so presentation.ts no longer shares a name with src/diagnostics/presentation.ts.
• Added test/login/badge/context.ts with the four module contexts built from BadgeTestFixtures, replacing the repeated 12- and 13-key Pick lists and the Generation/report types written in the status module.
• The entry's registrar imports now come after its other imports, BadgeTestFixtures sits right after the fixtures and harness, and the stray blank line in the imports is gone.
• The docs section now sits after the sibling test organization sections, describes the layout rather than the change, and no longer calls this the sole badge entry (badge.exhaustive.test.ts is a separate entry).
• Merged main; the docs conflicts are resolved by keeping every main section and adding this one before "Login wire contracts".

Verification

• Vitest JSON reporter test IDs for src/login/badge.test.ts: 134, identical to main and all passing.
• Inlining the registrars back gives the main file, apart from the moved imports and the exported type.
• oxfmt --check, oxlint apps/extension, rumdl check docs/development.md, mise run //apps/extension:typecheck pass.

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