Interestingly, Safari at least used to support the first - if you type something into the bar on a site which advertises opensearch, those results will be incorporated into the displayed list.
For the global behavior, the guidance seems to have shifted from opensearch plugins to web extensions. Safari does not seem to support any of the search engine metadata fields currently.
Chrome uses HTML5 search inputs: https://martin-thoma.com/search-engine-autodiscovery/
Make search engines (appearing in that list) installable as apps.
Have search engine websites include some metadata about the API. When this metadata block is present, add a "make this site the default search engine" to the Safari menu.
Allow the user to enter an arbitrary URL with some substitutions.
Safari however does not currently support new search engines via web extension metadata.