[DOCS] Add overlays for connector API examples

[lcawl]

Relates to #2482

This PR adds the connector API examples to the OpenAPI document via overlays.
It also fixes some duplicate labels and incorrect entries (e.g. cat threadpool should not exist in Serverless) in the overlay files.

[github-actions]

The backport to 8.x failed:

The process '/usr/bin/git' failed with exit code 1

To backport manually, run these commands in your terminal:

# Fetch latest updates from GitHub
git fetch
# Create a new working tree
git worktree add .worktrees/backport-8.x 8.x
# Navigate to the new working tree
cd .worktrees/backport-8.x
# Create a new branch
git switch --create backport-3644-to-8.x
# Cherry-pick the merged commit of this pull request and resolve the conflicts
git cherry-pick -x --mainline 1 30825df3110c3b4f6866f15bc322f60c21ae600b
# Push it to GitHub
git push --set-upstream origin backport-3644-to-8.x
# Go back to the original working tree
cd ../..
# Delete the working tree
git worktree remove .worktrees/backport-8.x

Then, create a pull request where the base branch is 8.x and the compare/head branch is backport-3644-to-8.x.

[lcawl]

💚 All backports created successfully

Status	Branch	Result
✅	8.x

Questions ?

Please refer to the Backport tool documentation