WYSIWYG – Getting the “link” button from HTML mode in Visual mode

Sorry, but no. 🙂 You can’t simply use the html-button in the wysiwyg-editor. Both are different programms and have nothing in common except for their location. You’d have to unregister the current buttons, write code for a new one (that fits your likings) and then add this one. You’ll find hundreds of tutorials for adding buttons to tinyMCE both on google, as here in WPSE.