I have heard of many people who claim that the Mishnah Berura doesnt hold of a Kli Shlishi based on Mishnah Berura 318:39. But it seems to me that the Mishnah Berura does hold of Kli Shlishi in 318:45. But when I say this, people say, why then do we need the Eitza of Tea Essence. So I respond that maybe the Mishna Berura does hold of Kli Shlishi, but is more confident with the tea essence idea. But MeIkar HaDin he does hold of Kli Shlishi. Does this make sense?
It does make sense as a possibility. Mishnah Berurah (318:45, 47) clearly writes that there is a distinction between a keli sheini and a keli shelishi, unlike Chazon Ish (Orach Chaim 52:19) who makes no distinction between them. Another possibility is that Mishnah Berurah is concerned specifically for tea, because we see that tea gets cooked even in a keli shelishi, a claim expressed by Aruch Hashulchan and Shemiras Shabbos Kehilchasah (1:57). See, however, Iggros Moshe (Orach Chaim 4:74, sec. 15) and Yabia Omer (7:40), who defer the argument, based on the fact that tea is ‘cooked’ even in cold water.