Answer:
The correct term is just "platypus", and there is just one species (Ornithorhynchus anatinus) which is the single remaining representative of its taxonomical family (Ornithorhynchidae) and genus (Ornithorhynchus), though a number of related species have been found in the fossil record.
Originally, the platypus was described as Ornithorhynchus paradoxus, in either 1799 or 1800, by German anatomist Johann Blumenbach. This was done by an independent examination, and the name was not official.
Australians do not refer to this creature as a "duckbilled platypus"; that is just a name that was carried over from colonial times, when the English first discovered the platypus, and which non-Australians continue to use.