Reset the plugin-update notifiers live after updates run

[epeicher]

Closes #296

What it does

After updating plugins (or themes) inside the desktop shell, the two update notifiers now reset live, with no hard refresh:

• the admin-bar circle-arrows count at the top-left (#wp-admin-bar-updates) — the element reported in Plugin update notifier does not reset #296,
• the dock Plugins badge.

Both flows are covered: per-plugin "update now" links on plugins.php (Core's shiny AJAX updater) and the bulk "Update Plugins" run on the Updates screen. When the count reaches zero the admin-bar node disappears, exactly as a fresh server render would; if updates appear later on a shell that booted clean, the node is re-created in Core's slot order.

Rationale

Two gaps compounded:

1. The chromeless bridge emits a fresh menu payload only on a full page load of plugins.php / themes.php / etc. Shiny updates run over AJAX with no navigation, so the shell never learned the counts changed — the dock badge went stale.
2. Nothing ever repainted #wp-admin-bar-updates: it is static server HTML rendered at shell boot, so even a fresh payload could not fix it.

A third, related gap: the bulk updater nests a progress iframe (update-core.php → update.php?action=update-selected) whose iframe_footer() fires admin_footer after the upgrades ran — a perfectly fresh payload — but posted it to window.parent, i.e. the intermediate update-core.php page, which has no listener.

Implementation

Follows the existing live-refresh pattern (payload key + shell applier), no new mechanism:

• includes/render/chromeless-bridge.php — a shiny-update watcher binds the jQuery events updates.js triggers on document (wp-plugin-update-success / -error, wp-plugin-delete-success, theme counterparts) and posts a payload-less desktop-mode-updates-changed nudge once wp.updates.queue is drained. Error events are included deliberately: wp_ajax_update_plugin calls wp_update_plugins() up front, which can mutate the transient even when the upgrade fails. The payload/signature emit now targets the top window instead of the immediate parent — identical for normal window iframes, and it lets the nested bulk-updater payload reach the shell.
• includes/core/payload.php — the menu payload gains updateCounts (total, locale-formatted count, translated screen-reader text, update-core.php URL) from capability-aware wp_get_update_data(), prebuilt server-side so the client repaint stays locale-correct.
• src/boot/menu-refresh.ts — handles the nudge with a 600 ms trailing debounce plus in-flight coalescing (a nudge landing mid-probe queues exactly one follow-up probe), then spends one existing refreshMenu() probe.
• src/admin-bar-updates.ts (new, wired from src/menu-refresh-apply.ts) — mirrors updateCounts onto the admin-bar node: label + screen-reader text, hidden at zero, re-created mirroring Core's wp_admin_bar_updates_menu() markup when absent. Missing key (older payload) means "no change". The dock badge needs no new code — the probe's payload already recomputes it.

Docs updated in the same change: docs/javascript-reference.md (payload shape + new message), docs/api-index.md, and the payload key listing in AGENTS.md.

Testing instructions

npm run test:js    # includes 8 new cases for the admin-bar repaint + 2 payload-contract cases
npm run test:php -- --filter test_payload_carries_update_counts

Manual repro (mirrors the issue):

1. Install outdated plugins, e.g. wp plugin install classic-editor --version=1.6.5 and wp plugin install akismet --version=5.3, then wp eval 'wp_update_plugins();'.
2. Load the desktop shell — top-left shows the circle-arrows count, the dock Plugins tile shows a badge.
3. Update the plugins from the Plugins window ("update now" links) — the count steps down live and both notifiers disappear at zero.
4. Repeat via the Updates screen with "Update Plugins" (bulk) — same live reset when the run completes.

Verified end-to-end on wp-env: counter stepped 3→2→1→hidden with per-plugin updates, and the bulk run cleared both notifiers the moment "All updates have been completed" appeared. Full suites green: 2178 vitest tests, 1333 PHPUnit tests, typecheck, ESLint, PHPCS (no new violations).