Sort of - there were different terms adopted by different countries. Submachine gun was an American term, while machine pistol was used by the Germans, Soviets (as пистолет-пулемёт) and French (as Pistolet Mitrailleur). The Brits had their own preferred term, machine carbine. The confusing part is that later on, machine pistol was also adopted as a term in American usage, except to refer to a specific subtype of submachine gun (in the most literal sense, just pistols that can shoot in full-auto, like the Stechkin, but also more broadly, really compact SMGs that have a form factor comparable to a pistol).

The StG-44 originally being called an MP was just a trick by the designers, since Hitler was initially opposed to it, so in order to keep developing it they just pretended that they were working on a regular SMG to replace the MP-40, and once it was accepted into service they redesignated it to Sturmgewehr. Interestingly, before the Maschinenpistole designation trick, the earliest prototypes were actually called Maschinenkarabiner - the same term the British used (except the Brits used it for guns in pistol-calibers, while the Germans already had Maschinenpistole for them, so Maschinenkarabiner now referred to automatic stuff in a bigger caliber).

So yeah, there's definitely a bunch of confusing stuff about the terminology - part of the problem is that the focus on the "machine" part made sense when the vast majority of weapons were bolt-action rifles, so having something self-loading was a big deal, and it had to be distinguished from regular guns. But when the vast majority of infantry weapons become self-loading, technically they're all some kind of machine something.