Dabigguy- You are WAY overthinking this. DATY is just oral sex on a woman.
The answer to your above question is that it can be either.
If you are a man (as your name suggests) having a date with a Sheris courtesan, well then there is only one “Y” in the room to be dined on, isn’t there.
If you are a woman, then you’d need to decide with your chosen companion who is giving and who is taking…or if you take turns!
Best of luck!
YB