Tolkein's letter to Milton Waldman published in the second edition of The Silmarillion has a little more to say.
After Melkor was overthrown and cast out, many of the Elves listened to and made common cause with his former lieutenant, Sauron. At Eregion, they made Rings of Power (number unspecified). The Elves made Three supremely beautiful and powerful rings, without Sauron's help, though perhaps not free of his inspiration ("almost solely of their own imagination, and directed to the preservation of beauty"). Secretly, in the subterranean fire, in his own Black Land, Sauron made the One Ruling Ring. However, the moment he "assumed the One" (wore it? used it?), the Elves were aware of him, and hid the Three, so Sauron never discovered them or sullied them. So far so good. But what of the Seven and the Nine?
In the resulting war between Sauron and the Elves Middle-earth, especially in the west, was further ruined. Erigion was captured and destroyed, and Sauron seized many Rings of Power. These he gave, for their ultimate corruption and enslavement, to those who would accept them (out of ambition or greed).
Sauron seized "many" Rings of Power. We know he distributed sixteen to Men and Dwarfs. The phrasing implies that he distributed all he had ("These he gave"), so perhaps sixteen were all he got. Or perhaps he seized sixteen Rings of Power and a number of lesser rings. But the wording implies that there were other Rings of Power he didn't seize, others beside the Three.
The Three and the One were special. The Three were the works of the Elves only, and were of special beauty and with a specific power directed to the preservation of beauty. The One was the work of Sauron only, and was designed to rule. The others (at least sixteen, by the phrasing likely more), were collaborations.
However, the letter mentions no distinction between the twenty (or more) "great rings" and other "lesser rings".