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)?
אֶחָד — "Echad" means "the only one" or "alone", not "one". Shema Israel: YHVH our God! YHVH is the only one! Or "Shema Israel! YHVH is our God! YHVH alone!". You cannot count God. As well as we cannot count water, fire, air, light. So He created light in the first and "only day" (Yom Echad) of creation. This is why today we say "Yom Rishon", because it is no longer Yom Echad "the only day of creation", is the first of the week.
On day one, day one was not the first in a series of days, it was the one and only:
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).