The Scrum Master role is many things to many people.
These days Scrum is sold to companies in the form of a (highly paid) consultant who will come in for a fix period, setup software, train staff and be the Scrum Master. Kick starting the process and moving people into "Agile"
Of course, when the consultant leaves, the Scrum Master has left. If the consultant did his job well, logically there should be no need for a Scrum Master anymore.
In the old days the scrum master would be the Team Lead, or the Project Manager. They would have a normal job to do as well as manage the Scrum.
But perhaps because the consultant was so highly paid, there is a reluctance to let the role fall into the hands of these lowly peons. They might leave and become Scrum Consultants!
Sometimes companies will have floating Scrum Masters, who hop between teams troubleshooting issues where they occur
Sometimes there will be a role above Project Manager, a Product Manager or Project Architect who will take over the Scrum Master role after the consultant leaves.
Unfortunately the role title "Scrum Master" was invented to appeal to techies in the days when it was fashionable to call yourself a "Web Master" and invent funny titles. If you are in management, your job title is a bit more important to you. So "Scrum Master" tends to be swept under the carpet in favour of "Manager of X" or "Vice President of Y"
In summary there is no established term for Scrum Mastering. You are a "Manager" running a team or a "Consultant" selling Scrum