The link is correct. The purpose of that control is to provide a URL to the doc page with all the parameters prefilled. This is what most users expect and is similar to how that link control works on the rest of Stack Exchange.
To get a direct link to the API, click on the link next to the Run button:
This has a URL like:
Yes, an edit will fix that.
This is because posts are stored as HTML. At the time the answer was submitted the build wasn't live. Therefor the tag wasn't a mod-only tag yet. It got no special handling. An edit will re-render the HTML and store a new version of the HTML. As the tag has now become mod-only it will render as red.
From the docs:
Searches a site for any questions which fit the given criteria.
The searches on the API aren't guaranteed to exactly replicate the current on-site search, so this behaviour is expected. This may change in some future release of the API, but the current behaviour is closer to the old on-site search approach, which only ever ...