Yes, the saving it as an alternate style only applies if instead of downloading the actual stencil from Mr. Hawk's studio, you instead download a lot on which several items have been placed. If you download and install a lot, then you need to save each stencil, and then save your game. After that, you can uninstall the lot if you like, and the stencils will still be there. If you download and install the actual stencil, then you don't need to do anything else.
I kind of would prefer the lot option, but there isn't a lot that actually has all of them. Both of the lots that I linked to only have some of them.
