Humm, this is an interesting conundrum. Would we consider a Scrum Master that coaches more than a team a part-time Scrum Master?
In my opinion, No. The only world where I see this work is if the team also works part-time.
Working part-time implies that you will not be available for the full days of work in your company. If your team is there for the complete working day and they have an issue that needs their Scrum Master, how should they proceed? Save the issue to be resolved the next day when the Scrum Master comes back to work? Pretend the issue doesn’t exist until the Scrum Master “shift” starts?
Scrum Master, which coaches more than one team, is available for the full day. If a team is in need, they know where to find the SM. It is highly unlikely that two teams will face problems at the exact same time. But it is highly likely that they will face a problem if you are not around.
Have you ever worked as a part-time Scrum Master? If so, let me know your experience.