Havent played minecraft via xenia so cant comment there, but I did play minecraft on xbox360 and I recall it having input lag, about half a second to a second.
Bit of a tangent, but I also played on ps4 and the switch and it has input lag on those too. I think minecraft was just not ported well to consoles if I am being honest.
I only remember it when in the world itself. It may have been in the menu too, I just dont quite remember