Here’s my thinking…
King of the Trident is probably the oldest title, a bit like Kings of Winter. It represents a claim to the defensible interior of the Riverlands, the part that would have been the hardest for the Westerlands or the Vale or the Reach etc. to conquer.
King of the Rivers and Hills is a more expansive title. The Hills portion represents a claim to the hill country that runs from Pinkmaiden to Harrenhal, the portion I’ve described as the “southern Riverlands,” which is the vulnerable underbelly of the kingdom because it’s not sheltered behind the riverrine walls of the Trident. The Rivers portion is not only a repetition of the claim to the Trident, but also a maximalist claim to all the rivers of the Kingdom, including the God’s Eye River and the Blackwater Rush, and thus might well represent a claim on the northern Crownlands, since King’s Landing and Rosby (and Duskendale, conquered by Benedict II, who may have been the first to use that title) are quite close to the Rush, Cracklaw Point is quite close to Maidenpool and borders on the Trident as it opens into the Bay of Crabs, etc.
River King is the colloquial term. If the two above are the titles that a Justman monarch’s court would use, I imagine River King would be the term used by the smallfolk or by foreigners to differentiate these kings from the many other kings of Westeros.

