Extensions: advertised-extensions toggle in Server Settings (Phase 2 of #1633) (#1743)

* Extensions: advertised-extensions toggle in Server Settings (Phase 2 of #1633)

Adds the debugging knob from #1633: users choose which extensions the
Inspector advertises to a server, and a server may register different tools
in response. Builds on Phase 1's registry + advertisedExtensions option (#1738).

Settings + persistence (core):
- InspectorServerSettings.advertisedExtensions + StoredMCPServer field, wired
  through serverList's read/write converters and INSPECTOR_FIELD_KEY_MAP
  (omit-when-empty, byte-stable round-trip).
- validateSettings (remote PUT route): isBooleanRecord guard + carry a
  non-empty map; unknown/malformed rejected.
- App.tsx threads savedSettings.advertisedExtensions into the client options.

UI:
- ServerSettingsForm renders an "Advertised Extensions" group of checkboxes
  driven by ADVERTISABLE_EXTENSIONS; a per-server override wins over the
  registry default. Modal folds a toggle into settings.advertisedExtensions.

Test server + proof:
- composable-test-server extensionGatedTools: a tool registered disabled and
  enabled on `initialized` iff the client declared the mapped extension
  (getClientCapabilities). Legacy stateful leg. Threaded through
  load-config/resolve-config; advertised-extensions-http.json showcase + README.
- Integration test connects legacy with/without the tasks extension advertised
  and asserts the gated tool appears/disappears in tools/list — the acceptance
  criterion, proven end to end, not asserted.

Tests: serverList round-trip (non-empty/absent/empty), ServerSettingsForm
toggle rows, ServerSettingsModal fold, validateSettings accept/reject, and the
integration gating test. npm run ci green (per-file ≥90 gate held).

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01XNdjEPKLG637X8YmhDiEk5

* Extensions: add #1739 proof screenshots (advertised-extensions toggle)

End-to-end smoke test in the web client against the legacy
advertised-extensions-http.json test server (port 3220): connecting with the
Tasks extension advertised (default) lists echo + get_weather, unchecking the
Advertised Extensions → Tasks toggle and reconnecting lists only echo — the
server's tool registration changing on the client-declared extension.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01XNdjEPKLG637X8YmhDiEk5

* Extensions: reconverge advertisedExtensions to no-override on default (review)

Address Phase 2 review observation #1: toggling an extension back to its
registry default now deletes the override key (and drops the whole map to
undefined when it empties), so the on-disk `advertisedExtensions` reconverges
to the byte-stable "no override" state instead of persisting a redundant
`{ key: <default> }`. Matches the omit-when-default policy the other settings
use. Only a value that differs from the default is persisted.

Adds a modal test asserting the reconvergence (re-checking the sole disabling
override drops advertisedExtensions to undefined).

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01XNdjEPKLG637X8YmhDiEk5

---------

Co-authored-by: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

pr-screenshots/README.md

@@ -1,3 +1,39 @@
+# Advertised-extensions toggle (#1739) — proof screenshots
+
+End-to-end verification of the advertised-extensions debugging knob against a
+real legacy HTTP test server (`test-servers/configs/advertised-extensions-http.json`
+on port 3220), driven in a real browser through the web client's remote-proxy
+transport. The server registers `echo` (always) and `get_weather` **gated on the
+`io.modelcontextprotocol/tasks` extension** (`extensionGatedTools`): the tool is
+enabled on `notifications/initialized` only when the connected client declared
+that extension.
+
+![Advertised Extensions — Tasks advertised (default)](advertised-ext-settings-tasks-on.png)
+
+**Server Settings → Advertised Extensions** (the new section). With no override,
+**Tasks (io.modelcontextprotocol/tasks)** is checked — the registry default —
+so the Inspector advertises it in its client capabilities.
+
+![Tools with Tasks advertised](advertised-ext-tools-tasks-on.png)
+
+Connected with Tasks advertised. The server sees the declared extension on
+`initialized` and enables the gated tool, so `tools/list` returns **both `echo`
+and `get_weather`**.
+
+![Advertised Extensions — Tasks unchecked](advertised-ext-settings-tasks-off.png)
+
+Unchecking **Tasks** writes the per-server `advertisedExtensions` override. The
+change takes effect on the next connect (as the section note says).
+
+![Tools with Tasks not advertised](advertised-ext-tools-tasks-off.png)
+
+After reconnecting, the client advertises no extensions, so the server never
+enables the gated tool — `tools/list` now returns **only `echo`**. The server's
+tool registration demonstrably changed based on what the client advertised,
+exactly the acceptance criterion for #1739.
+
+---
+
 # x-mcp-header Tools tooling (#1632) — proof screenshots
 
 End-to-end verification of the SEP-2243 `x-mcp-header` Tools tooling against a