I have heard multiple names for water-source heat pumps. What's the difference between something called a water-to-water heat pump, and a seawater heat pump? Are they just different names for the same thing?
They can be the same thing; the terms have overlapping meanings.
A seawater heat pump is a specific type of water-source heat pump that uses seawater. Water-source heat pumps can also use fresh water, e.g. from a lake or river.
The source is the first part of the meaning of "water-to-water". The second part means that the destination for the heat is water, most likely in a "wet" central heating system with pipes and radiators. One could also have a water-to-air heat pump, which would use water as the source but deliver the heat through hot air from a wall unit.