Scrum Master as Coach & Process Referee: Teachable Moments
Is it the responsibility of the Scrum master to intervene and interrupt a team member when he/she starts telling about points beyond those 3 that I mentioned?
Yes, it is. The Scrum Master is a servant-leader whose primary responsibilities are to act as both a coach and a process referee. The Scrum Guide specifically calls out this responsibility by making the Scrum Master responsible for proactively facilitating Scrum events such as the Daily Scrum. It makes this proactive, discretionary aspect clear when it says (emphasis mine):
Facilitating Scrum events as requested or needed
The section on the Daily Scrum goes on to say (emphasis mine):
The structure of the meeting is set by the Development Team and can be conducted in different ways if it focuses on progress toward the Sprint Goal. Some Development Teams will use questions, some will be more discussion based...
The Scrum Master ensures that the Development Team has the meeting, but the Development Team is responsible for conducting the Daily Scrum. The Scrum Master teaches the Development Team to keep the Daily Scrum within the 15-minute time-box.
The Daily Scrum is an internal meeting for the Development Team. If others are present, the Scrum Master ensures that they do not disrupt the meeting.
While it might be tempting to treat the second paragraph above as a prohibition against any kind of Scrum Master participation in the Daily Scrum, the contextual intent is to to prevent the meeting from turning into a status pull from the Scrum Master (or anyone else). The Daily Scrum is a coordination meeting, not a report to an authority figure.
In fulfilling the Scrum Master's role as coach and process referee, a Scrum Master is obligated to point out when the framework isn't being followed, or when the team is missing an opportunity to leverage the framework to best advantage. This is subtly different from the Scrum Master running the meeting. If you think of it more as a teachable moment, rather than a power play, then you're on the right track!