Move JSDoc examples to type-checked .examples.ts files

Extract inline code examples from JSDoc comments into companion
`.examples.ts`/`.examples.tsx` files. Examples are now:

- Type-checked by wrapping them in functions
- Referenced via `{@includeCode ./file.examples.ts#regionName}` tags
- Organized using `//#region` and `//#endregion` markers

This keeps the main source files cleaner while ensuring examples
compile correctly. Updated TypeDoc config with `exampleTag: false`
to support title text on `@example` tags.

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

typedoc.config.mjs

@@ -17,6 +17,9 @@ const config = {
   gitRevision: "main",
   excludePrivate: true,
   excludeInternal: false,
+  jsDocCompatibility: {
+    exampleTag: false,
+  },
   categorizeByGroup: true,
   navigationLinks: {
     GitHub: "https://github.com/modelcontextprotocol/ext-apps",