What term or phrase means a person's "manner of speaking" in the general sense?

Elocution is defined as "a particular style of speaking" (here) and "a person's manner of speaking or reading aloud in public" (here).

Given that both definitions limit application to speaking aloud, it seems to fit your bill. Further, it is more holistic (that is, encompassing) than words like cadence or prosody which relate only to the formal properties of speech.

'Voice' could be used to describe one's manner of speech, though it would take some setup to ensure the desired meaning, and not the more obvious one, is properly established.

idiolect - the speech habits peculiar to a particular person or idiom - the style of expression in writing, speech, or music that is typical of a particular period, person, or group (please note that 'idiom' has multiple meanings)