The second perek of Massekhet Sota teaches about the preparations for the ceremony itself. The Mishna on our daf teaches about how the “bitter water” is prepared. The kohen would bring an earthenware peyalei, and would fill it with half a log of water from the kiyor – the Temple washbasin. Rabbi Yehuda requires only a revi’it, one quarter of a log. Based on the passage in Bamidbar (5:17), dirt would be taken from the floor of the mishkan – or from under the stones of the Temple floor – to be placed in the water.
The peyalei – or the Greek libation vessel, the phiale – was a utensil used for cooking and drinking. The peyalei could be made out of different materials, e.g. from earthenware or metal, and it was apparently shaped like a shallow pan (the Greek translation of the word ke-ara that appears as a utensil used in the mishkan is phiale).
One question that is raised about our Mishna is why a Greek word is used to describe this utensil, rather than using a Hebrew word like kos (cup) or kedera (bowl). The Meiri points out that the Targum of the expression kuba’at kos ha-tarela – “the cup of staggering” – in Yeshayahu (51:17) is, in fact, peyalei, which hints to the cup from which the sota drinks, as well.
The word peyalei is found many times in the Targum that originated in the Land of Israel as the translation of the word sefel (a bowl) – see, for example, Shoftim 5:25 in the Targum Yonatan.
Perhaps the simplest explanation of why the Mishna chose to use this term is because it refers to a very specific type of utensil, which could not be expressed with the use of a generic term like “cup” or “bowl.”