ci: Restore lychee link check. Only validate internal links (#32463)

Follow-up to: https://github.com/zed-industries/zed/pull/32460
Follow-up to: https://github.com/zed-industries/zed/pull/30844

Release Notes:

- N/A

.github/actions/build_docs/action.yml

@@ -19,6 +19,12 @@ runs:
       shell: bash -euxo pipefail {0}
       run: ./script/linux
 
+    - name: Check for broken links
+      uses: lycheeverse/lychee-action@82202e5e9c2f4ef1a55a3d02563e1cb6041e5332 # v2.4.1
+      with:
+        args: --no-progress --exclude '^http' './docs/src/**/*'
+        fail: true
+
     - name: Build book
       shell: bash -euxo pipefail {0}
       run: |