It is (AFAIK) more of a military-specific term, and means being disciplined rather than fired. I believe it stems from a "breach of discipline".

Is it a synonym for, or just a milder version of, what Americans refer to as "court-martial"?

I would imagine that it covers the sort of disciplinary offences where one's commanding officer can impose some form of punishment or official reprimand without there needing to be a court martial. The US equivalent would be, I think, a Nonjudicial Punishment or Article 15.

Purely as an FYI, what Americans would refer to as a court-martial would also be a court-martial in the UK.

I would think it'd be closer to a mast/NJP (non-judicial punishment) where you basically get yelled at, get some sort of grunt duty (cleaning the latrine for a month?), maybe have a note in your file if the CO deems it severe enough.

No, it's basically a dressing down and formal disciplinary process.