Maxperson
Morkus from Orkus
You're both right. This is what it says.If they had to take an Oath at level 1, absolutely. Except, they don't have to make that choice...
Granted, yes, in one's individual campaign, you can explain it thusly, and in fact, force players to think about their subclass choices at level 1- which really, they should, but since the game doesn't suggest this, it does come off as a bit odd.
"When you reach 3rd level, you swear the oath that binds you as a paladin forever. Up to this time you have been in a preparatory stage, committed to the path but not yet sworn to it."
While you are correct in saying that they have not yet sworn the oath, they have already chosen it and have been committed to that path, which makes @Crimson Longinus also right in saying that this is likely what gives them the first two levels of powers.