I don't think you've scratched the crystal. And I don't think you will scratch the crystal no matter what you do to remove the blemishes.... I have tested a sapphire crystal with a metal dremel pad bit (and no pad attached) at 30K RPM. No scratches. I think the only way that you might possibly scratch a sapphire crystal with a dremel is if you let the crystal heat up significantly while polishing.
The other possibility is that your crystal is not sapphire.
Regarding the TZ article... He scratched true sapphire with a ceramic tile? Not likely. But regarding sapphire polishing, yes diamond paste does work. The issue is that you must start with a size as large as your scratches and move down in size to achieve the final polish. So each situation is different, and what worked for the TZ guy might not work for other scratches. The TZ guy found something that did the trick for 8 GBP per tube, but that's quite a good price. tedpella.com is a source for these pastes that I've talked to, but the least expensive route that might work for a variety of scratches would be the 895-20 set of monocrystalline diamond polishing compounds for $67.60 applied with 813-10 3M Scotch-Brite High Performance Cloth for $17.75. A new crystal is probably cheaper.