A sui-teki [水滴] is a water-dropper. It was used to control the amount of water added to an ink-stone (suzuri [硯]), when preparing writing ink. Some sui-teki have an extremely small hole in the spout, so that the water literally is dispensed in droplets, while others (such as the one shown above) are shaped more like ordinary cruets. During the sixteenth century, imported continental sui-teki were sometimes used as chaire, since they were often made of the same sort of brown-glazed pottery as the small jars more commonly used chaire.
This name, however, might be a corruption of (or a conflation with)* the name su-teki [酢滴], which means a vinegar dropper, since the first known container of this type (which possibly made its first appearance as a chaire during the second half of the fifteenth century; though unambiguous references are not found until the second or third decade of the sixteenth century) was apparently one of a set of four similarly-shaped vessels (that had been imported from the continent) which were originally made as condiment containers for restaurants to display on their dinner-tables†. The four were the su-teki [酢滴], the yu-teki [油滴], the te-kame [手瓶], and the tsuru-tsuke [弦付]‡.
†We must remember that the Shukō chawan, and several of the other bowls that were treasured by chajin during his period (all of poor-quality celadon-ware), were originally made as a noodle-bowls for use by road-side food booths. The original set of four containers (su-teki [酢滴], yu-teki [油滴], te-kame [手瓶], and tsuru-tsuki [弦付]) likely came from the same kind of restaurant.
These things were usually originally brought from China by monks who were essentially forced, by convention, to bring souvenirs for the people who had financed their study-trips through donations (even though they rarely collected more money than would be necessary for the transportation to and from China, expecting to rely on begging for their day-to-day necessities) - and (in Korea and Japan) these objects were treasured by the recipients accordingly. So while such pieces were considered to be of low quality in China (but the same must be said of the Chinese "chaire," which were either made as medicine jars, or as containers in which hair pomades or single-servings of medicinal-wines were sold as souvenirs from famous scenic places, and considered to be of no value once the contents had been consumed), they were treasured in Korea, and even more so in Japan, because they were technologically advanced, compared with the contemporary indigenous pottery traditions in these other countries.
†Which suggests that the changes found here were the result of clerical errors, rather than an editorial attempt to change the meaning of the original to conform with contemporaneous practices.
In fact, it seems that the way to handle the sui-teki no chaire was pretty much a mystery to the early Edo practitioners of chanoyu, which accounts for their misunderstanding of this passage. They understood saki to mean on the side of the sui-teki facing toward the person who will pick it up (which is the way such containers are oriented on the table when they contain soy sauce or some other liquid condiment), rather than on the far side (as Rikyū intended): this is how the deviation between the modern way that this tea container is used, and what Rikyū held to be the proper, orthodox, handling, arose.