Are verbs such as movere third conjugation because its thirt principle part is movi, not movui?
It's in third conjugation because the first "e" is short, not long. (No macron over that "e" or any other in the third conjugation.)
Look at habeo, habere (there should be a macron over the "e" in the first principal part and over the first "e" in the infinitive. This is second conjugation.
It also affects the pronunciation:
habere is pronounced ha-BAY-reh, while movere (no macron over any vowel!) is prnounced MO-ve-reh.