feat(notifications): extension-badge delivery channel

[JSONbored]

Parent roadmap: #525
Parent epic: #535

Goal

Delivery channel #1 — extension badge (pull-based): GET /v1/extension/notifications?since= + a log-row write on delivery.

Reuse

The extension API (#556).

Acceptance

A delivered notification appears in the poll endpoint exactly once.

Testing

npm run test:ci, 97%+ coverage. Public/private output boundary preserved; deterministic where applicable.

[JSONbored]

Verified shipped (2026-06-17 audit): extension-badge delivery channel implemented (resolveNotificationChannels → badge) + delivered, tested (mcp-notifications.test.ts). Closing (email channel remains in #570).