The above are mild abrasives, they do nothing to glass.
There are various polish thread on this forum, use the search function. Anyway:
Place the wet paper on a plastic support disc 5 - 8 cm diameter.
Observe from inner side that you are hitting to the correct spot, orthogonal to the scratch direction. You will see white streaks on the point you're abrading.
Check often and as soon the scratch is gone then move to 600, that's the finest grip that will have some effect. After that you'll need diamond paste.
You can't ruin a crystal unless you drop and chip it. Of course, on a faceted crystal, crisp edges cannot be preserved.
I've got a dremel with the small felt wheels, that should suffice? I will need to pick up some of that diamond paste. Like I said, I've been trying to use the two polished below, finishing with Peak Cream. I think starting at 1000 grit was my biggest problem. Will pick up some rougher grit SP and give it a whirl. Worst case scenario is a ruined crystal i know i can replace!
If the issue is the deep scratches, you can't get them off with 1000, start with 280. Abrading glass is not like working to paint or metal.
Then at the end you can't get good finishing with paper only, crystal will always appear kind of blear. Use diamond paste on a felt wheel, mounted on a rotary tool. Don't bother with paste below 20 grit. Both paste and the tool, in case you haven't it, are very cheap on AliX or similar sites.
I'm think that is more an artifact of the low res display of the 1000 model than a real problem. In my opinion in the picture above the relation between lines is good, only the instrument draws some point one pixel off due to to rounding algorithm.
In other words it makes you see things worst than they are.