shabbat - Where does the concept of a Kli Shlishi come from?

When it comes to cooking on Shabbos, most people hold that things that are not allowed to be put in a Kli Sheni (a vessel once removed from the vessel that was on the fire) because it is considered cooking are able to be put in a Kli Shlishi (a vessel once removed from the Kli Sheni).

(See here for a short summary of these laws).

The Mishna Berurah (318:47) quotes the Pri Megadim who differentiates between a Kli Shlishi over a Kli Sheni.

The Pri Megadim (Ashel Avraham 318:35) quotes the BaCH who says that there is no difference between a Kli Sheni and a Kli Shlishi, and even a Kli Revi'i is considered like a Kli Sheni. He then says that with regards to tea, When one pours from "ממים טמונים‬" (which I'm interpreting to mean A Kli Rishon that was removed from the fire and kept warm through Hatmana, not if it was still directly on the fire - please correct me if I'm wrong) into a Kli Sheni, and from there into smaller containers, one can be lenient.

However, the Chazon Ish ( Orach Chaim, end of Chapter 52) says that he did not find a source to differentiate between a Kli Sheni and a Kli Shlishi and as long as the contents of the Kli Sheni are hot, pouring it into a Kli Shlishi doesn't change anything.

What is the source for a Kli Shlishi? If the Chazon Ish couldn't find a source for the difference, where do the Poskim that do differentiate get it from?

Somebody once told me that a Kli Shlishi is not mentioned in the Gemara or Shulchan Aruch. Is this true? If so, when was the concept introduced?