Meaning of "grounded" in a uniform field
In my opinion, "grounded" means having the same potential as the potential at infinity,


This is a useful definition in a universe containing only a finite number of charged particles hanging in space, but it's not used much in practical engineering.

For example, in your example, there's a uniform field. If you go to infinity in one direction you will be at a very different potential than if you go to infinity in another direction (assuming you don't go in directions perpendicular to the field). So we really can't talk about the potential at infinity and a uniform field in all of space in the same example.

In this case, the author is telling you to take the grounded conductor as the reference potential, rather than a location at infinity. Just as a matter of facilitating communication (with whoever's grading your paper, for example) it's better to just use their convention than to make up your own. And in this case, the supplied convention is probably a good choice, given the difficulty I mentioned above.