Is what was uttered in a soliloquy, with the intention of facilitating a matter, by moving the lips without being certain of sound emission, considered a vow in both of the aforementioned cases?

This is not considered a vow (nadhr). A vow requires intention and verbalization with a صيغة (formula) of commitment, not merely a passing thought or an unexpressed intention. The Messenger of Allah, peace and blessings be upon him, said: "Indeed, Allah has pardoned for my Ummah whatever they whisper to themselves, as long as they do not act upon it or speak of it."