At least where I teach, nothing automatically triggers. I can’t imagine how that would work without lots of false accusations.
TurnItIn comes up with possibly plagiarized passages and possible sources, but context would tell you if it is or isn’t plagiarizism.
There are plenty of scenarios where an assignment that triggers lots of flags wouldn’t be plagiarized as well as scenarios where an assignment triggers nothing is obviously plagiarized.
My institution doesn’t want us handling these issues off-the-record. If a student plagiarizes, we report it and go through the procedure. We don’t work out arrangements on our own for students to make up for the cheating. There have been situations where a student seemed to perpetually work out ad hoc arrangements and skated by with lots of plagiarism.
This way it’s tracked. We have a pretty well-defined policy to determine the punishment — it’s a combination of the severity and frequency. A one off like the violation in the OP would just be a zero on the assignment and some training.
I don’t always bother asking the student for their side of the story. If the plagiarism is obvious and unambiguous, I’ll just send them a notice, give a zero, and report it. There’s no point in it, just a chance for them to make up some dumb story and debase themself. I encourage them to use the appeals process if they’re unhappy they got a zero for copy/pasting from Wikipedia.
A self-plagiarism case like this would be pretty straightforward, so I’d probably just notify and report. Unless it’s a major assignment or the student is a repeat offender, I wouldn’t expect major ramifications.
Self-plagiarism gets covered in our intro to being a college student class as well as first semester English. I suspect most institutions are similar. A returning student after many years would likely fall through the cracks and would be more likely to accidentally commit the offense. Our policy is that ignorance is no excuse, but first offenses on minor violations are mostly teaching moments rather than a chance for punishment.
And lulz at a board full of lawyers talking legal options because someone might be punished for violating a clearly defined policy. I’ve served on our appeals committee for a number of years. We’ve had a few lawyer-assisted appeals and it has never helped the student.