Since I don't eat either, I really don't know. But they probably do have to do something different to get those slices to a texture that the wrapping machine doesn't destroy them and they don't break up when unwrapped. So the two probably are different in some ways.
I suspect you did it the only way you could. Now I see why the grating was a PITA though. Seems you would need 3 hands, 2 to hold the grater and one to run the cheese across the surface while the grater wants to move in the same direction.
We have one like this:
A cheap $7, there are more expensive ones. You might be able to buy one of these at the grocery store or in the kitchen section of a hardware store.
Stand it in a big bowl and hold it down with one hand and hold the cheese with the other. You don't HAVE to use a bowl but I figure that way there is not need to make sure the counter is really clean

You could just do it onto a cutting board as well.
With a box grater you are pushing both the cheese and the grater down onto a solid surface so you don't have to fight against yourself. I think it would be substantially easier than the kind you have, plus the grating surface is probably twice the size of yours. It has small holes on one side, large on the other. Can't say I've ever tried to slice using the 3 wide cutters nor the really fine grater nibs on the side opposite those. I think that would be used for really hard cheese like Parmesan that you want fine like it comes out of a shaker.
https://www.amazon.com/FlyingColors-Stainless-Steel-Boxed-Grater/dp/B0188Y2UHU
Only maybe 2-3". Mr. NOAA missed by a fair bit. I'm not complaining though, the 30 mile trip down to the in-laws was a non issue.
Fortunately y'all warned me not to look and I didn't! Sounds really bad.