Adds external link to execute watch API and removes request examples

[kosabogi]

DO NOT MERGE BEFORE elastic/docs-content#1780

This PR adds an external link to the Execute a watch API and removes unnecessary/redundant Request examples.

Related issue: https://github.com/elastic/docs-projects/issues/302

[github-actions]

Following you can find the validation results for the APIs you have changed.

API	Status	Request	Response
watcher.execute_watch	🟢	7/7	7/7

You can validate these APIs yourself by using the make validate target.

[leemthompo]

LGTM 👍

[leemthompo]

Looks so good, I'm gonna approve it instead of commenting 😄

[github-actions]

The backport to 9.0 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-9.0 9.0
# Navigate to the new working tree
cd .worktrees/backport-9.0
# Create a new branch
git switch --create backport-4634-to-9.0
# Cherry-pick the merged commit of this pull request and resolve the conflicts
git cherry-pick -x --mainline 1 0ef8855d679f070ea18c6b14b243895ea349f3c5
# Push it to GitHub
git push --set-upstream origin backport-4634-to-9.0
# Go back to the original working tree
cd ../..
# Delete the working tree
git worktree remove .worktrees/backport-9.0

Then, create a pull request where the base branch is 9.0 and the compare/head branch is backport-4634-to-9.0.