It's clear to me that $J$ is a principal ideal in $F[X,Y]$ and also that it is not maximal. A given example is: $J \subset (J,Y^2) \subset F[X,Y]$.
My problem lies with the example. I don't see why it should be $(J,Y^2)$ to properly contain $J$. Wouldn't $J \subset (J,Y)$ be also correct?
Sure, that's also correct. There are many different ideals that properly contain $J$; both $(J,Y^2)$ and $(J,Y)$ are examples, as are $(J,y+1)$, $(J, y^7+y+2)$, and many others. I'm not sure why they chose to give you the example $(J,Y^2)$; there's nothing particularly special about it, and $(J,Y)$ is in some sense the "simplest" example.