Both a kli sheini and a kli rishon shelo al ha’aish are pots of hot water that will gradually cool down. Since it is difficult to distinguish between them, we require Tosafos’s help to properly understand the distinction. Although they look almost identical, a kli sheini has difanos mikareros (walls that cool down the product) while a kli rishon shelo al ha’aish has difanos michamemos (walls that maintain the heat of the product). An extended irui is none of the above, for the simple reason that the walls of this pot will not cool down. So long as the irui continues, there is a heat source that is preventing the kli from cooling. For this reason it is most similar to a kli rishon al ha’aish.
The Taz 92:30 says that a ladle placed into a pot of boiling water will also become a kli rishon, if left there until saturated with heat and the water in the ladle boils. Even though the ladle never came in direct contact with the fire, nevertheless it is considered a kli rishon al ha’aish since it is heated by a toldos ha’aish. In this same manner a pot which is constantly heated by an extended irui can be considered a kli rishon, and can be mivashel. By way of illustration, although the Gemara Shabbos says that beef does not cook unless it is placed in a kli rishon al ha’aish, one can easily cook beef, and this is regularly done, through the use of an extended irui. By circulating hot water around the jacket of a pot one can heat the pot. This pot will stay hot for as long as it takes to cook the meat.
The Taz’s chumra of many iruis (95:12) is referring to a platter that is used numerous times, but each time it is only used as a conventional kli sheni. Since each irui is only boleya a kdei klipa, without the chidush of the Taz, one would say that multiple uses do not cause a greater bliya. Obviously one cannot use this chumra to create a kula. However, this should not be confused with an extended irui which creates the equivalence of a kli rishon al ha’aish.