API for working with App

You are right, the x-callback-url are usually called from a higher level and not directly from the browser. For example, on iOS you can use them from the Shortcuts app, in the Shortcuts section of the community you can find a number of downloadable Shortcuts that can show you how to do this.

Other ways you are use them is through AppleScript, shell scripts, Automator actions or other 3rd party apps that implement the x-callback-url type callbacks like Drafts, Instapaper etc. See for more usage examples: http://x-callback-url.com