Is what was intended to be slaughtered for the poor as an expression of gratitude considered a vow, a charity, or an Uḍḥiyah (sacrificial animal), and is the slaughter that occurred later, after some of it was eaten and distributed to non-poor individuals, sufficient, and does a vow remain binding upon the questioner based on their previous intention?

There is no vow upon you, nor anything. This is because a vow is a verbal commitment. As for your commitment through mere intention, it is not considered a vow, due to the saying of the Messenger of Allah, peace and blessings be upon him: "Indeed, Allah has forgiven my Ummah for what whispers to their hearts or what they contemplate, as long as they do not act upon it or speak of it." The default is the acquittal of responsibility, and a Muslim is not obliged to fulfill a vow unless he is certain he uttered it. Mere doubt does not obligate it. So, your sacrifice was a past Udhiyah.