All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
The Ahaadeeth reported in relation to the reason for venerating the Day of ‘Aashooraa' and why the Prophet, sallallaahu ‘alayhi wa sallam, observed fasting on that day is that it was the day when Allaah, The Exalted, saved Moosa (Moses) and his people from Pharaoh as in the following Hadeeth:
Ibn ‘Abbaas narrated that the Prophet, sallallaahu ‘alayhi wa sallam, came to Madeenah and he found the Jews observing the fast on the Day of ‘Aashooraa'. He asked about it and they said: "It is a blessed day. It is the day on which Allaah saved the Children of Israel from their enemy so Moosa observed fast on it." Upon this, the Messenger of Allaah, sallallaahu ‘alayhi wa sallam, said: "I am closer to Moosa than you," and he fasted on that day and commanded (Muslims) to observe fast on it." [Al-Bukhari and Muslim]
Another narration of the Hadeeth reads: "and it is the day when the ark came to rest on Al-Judi (mountain), so both Nooh (Noah) and Moosa observed fast on that day to show gratitude to Allaah." [Ahmad] Shu‘ayb Al-Arnaa’oot classified it in his revision of the Musnad as weak.
As for the other reasons mentioned in the question for venerating the Day of ‘Aashooraa', we do not know of any valid evidence to support them.