Whilst I think you are right, being put on a PiP should still be seen as a big warning. Especially since there are many ways to help someone improve in performance. There is little need for formalization in a benign pip. The main reason a pip is formalized is to prepare for firing.

I don't think being put on a pip means firing is certain and imminent. I do think being put on a pip means firing is on the table and a serious threat. In that sense the analogy of 'putting a bullet in the chamber' is accurate if maybe a bit strong.