Rabbi Menachem Greenblatt (St. Louis, Missouri) mentioned at s'uda sh'lishis (Shabas parashas "Ki Setze" 5772) that the correct name for Sunday is echad b(a?)shabas, as is written in k'suvos and gitin (see Shulchan Aruch, Even Haezer 126:3), even though the other days are referred to with ordinals (sheni et al.), and wondered aloud in passing why we say in the introduction to the shir shel yom "hayom yom rishon" and not "echad". Anyone have a reason (preferably sourced)?
Echad, Shtayim, Shalosh (Cardinal numbers)
Rishon, Sheni, Shlishi (Ordinal numbers)
Ordinal makes more sense to me: You are ranking (enumerating) the days, not counting how many there are.
But Bereishis mixes the two types - which is explained here: http://www.ou.org/torah/tt/5769/bereishit69/aliya.htm
"The day is called YOM ECHAD (cardinal number) rather than RISHON (ordinal number), because RISHON has meaning only if there is a SHENI, which there wasn't yet."
When you are davening, the second day does already exist, so you use ordinal numbers (as would be expected).
A get is different because you say "On day one of the week", which sounds OK. In contrast with davening the translation would be "Today is one day of the week", which could mean any of them, and really only makes sense when there is just one day.
So it seems to me that when the grammar allows it you copy the Torah, but when it sounds awkward you change it, (since it's anyway not a direct quote).
On day one, day one was not the first in a series of days, it was the one and only:
Each day of creation is numbered. Yet there is discontinuity in the way the days are numbered. The verse says: "There is evening and morning, Day One." But the second day doesn't say "evening and morning, Day Two." Rather, it says "evening and morning, a second day." And the Torah continues with this pattern: "Evening and morning, a third day... a fourth day... a fifth day... the sixth day." Only on the first day does the text use a different form: not "first day," but "Day One" ("Yom Echad"). Many English translations make the mistake of writing "a first day." That's because editors want things to be nice and consistent. But they throw out the cosmic message in the text! Because there is a qualitative difference, as Nachmanides says, between "one" and "first." One is absolute; first is comparative.Nachmanides explains that on Day One, time was created. That's a phenomenal insight. Time was created. You can't grab time. You don't even see it. You can see space, you can see matter, you can feel energy, you can see light energy. I understand a creation there. But the creation of time? Eight hundred years ago, Nachmanides attained this insight from the Torah's use of the phrase, "Day One." And that's exactly what Einstein taught us in the Laws of Relativity: that there was a creation, not just of space and matter, but of time itself . . . Now if the Torah were seeing time from the days of Moses and Mount Sinai ― long after Adam ― the text would not have written Day One. Because by Sinai, hundreds of thousands of days already passed. There was a lot of time with which to compare Day One. Torah would have said "A First Day." By the second day of Genesis, the Bible says "a second day," because there was already the First Day with which to compare it. You could say on the second day, "what happened on the first day." But as Nahmanides pointed out, you could not say on the first day, "what happened on the first day" because "first" implies comparison ― an existing series. And there was no existing series. Day One was all there was.