About 10 years ago I read a history of the SOTC written by the son of an Orthodox priest. One possibility he offered was that the priests at the beginning of Christianity would bless the people in what is now considered the Latin way. The people, in return would bless themselves by mirroring the motions of the priest, i.e., when the priest moved his hand from left to right, the people would move their hands from right to left.
As for the “Latin way”, there are a number of possibilities, most having to do with warfare. While one might wear a sword on his back in a break-away sheath, most of the time the sword would be worn on the left side so your right hand could pull it out. Thus, the Latin way imitated a warrior drawing his sword to do battle with the Evil One. Or, if you think of the SOTC as a shield, your action is pulling your shield across your front to defend you.
Two more things to remember about the Sign Of The Cross. First is that it was responsible for the greatest persecution in history, that of Diocletian. The Emperor would go to see the pagan seers once a year to learn what to expect for the coming year. One year, Diocletian brought along some of his guards who were Christian. Since they were in the presence of pagan magic, the guards crossed themselves for protection (probably using their thumbs on their foreheads.) This immediately drove the spirits away and the Emperor was left without his prophecies for the year, and that was the justification for the persecution.
Second is that Ezekiel 9:4 calls for the foreheads of the faithful in Jerusalem to be marked. While many modern translations transliterate the mark as an “X”, it is actually the Hebrew letter “taw” or “tav”. At the time of Ezekiel, it was written as a vertical cross, like a “t”. It is the last letter of the Hebrew alphabet, and when Jesus in Revelation calls himself “The Alpha and Omega”, that’s just for the Greek translators. What He said in reality was “I am the Aleph and the Tav”, identifying Himself with the cross.
FWIW, I frequently use the Eastern form when crossing myself at communion, in honor of my Russian-born wife (May God Keep Her Close.)