Why is gnocchi a dumpling and not a noodle?
The primary difference between a dumpling and a noodle, besides shape, is leavening.  Dumplings usually have either egg or baking powder to make them lighter.

There's considerable bleedover in terminology here. It's reminiscent of the difficulty of defining "chowder". You're always finding a counterexample. I wouldn't be surprised if someone came up with something called gnocchi, gnochs, gnocchetti, that has no leavening. I can think of two myself.

Gnocchetti alla romana are semolina "dumplings" sometimes leavened with egg, particularly when they contain spinach, but more often not. Gnocchetti sardi are dry pasta shapes that kind of look like thin potato gnocchi.