It really is. Great find, Handy! 
I’ll just chime in with a couple of cents. Long ago, in a life far, far away, I went to the top dog grooming school in the country (at least, at that time…I haven’t kept up), although I’m not currently doing grooming. We were taught the method shown on the web page Handy found (except for the “drape yourself over the dog” part…looks weird, but may be good for home groomers), as well as the electric method. The electric method is more dangerous to the dog than the manual cutters. This might seem counter-intuitive, as it seems more like the natural wear an outdoor dog’s nails are exposed to but, because of the high-speed nature of the tools involved, the similarity is more conceptual than factual.
Nerves damaged by grinding heal slower, more painfully, and less fully (by which I mean the pain may not entirely diminish). Grinding can also cause the fibers of the nail to separate enough for fungus to gain a (pardon the pun) foothold. That would be bad. This is usually the result of too much pressure and too little speed or too coarse of a grit. It is also possible to “melt”, burn or scorch the nail with an improperly used grinder. That’s usually due to too high a speed, plus too much pressure. This is also extremely painful.
In spite of all that, I’m not trying to talk you out of using the grinder, Padeye. If your spotted buddy really was traumatised by the PetSmart experience, it may be better to use a method he doesn’t associate with that experience. Just be very, very careful. Dremels run faster (higher RPMs) than real nailer grinders, so be especially careful not to overheat the nails, and use gentle to moderate pressure. Don’t keep the grinder on the nail too long at one time, either. You want to hear “zzzt, zzzt, zzzt”, not “zzzzzzzzzzzzzzt”. If your Dremel has variable speeds, use the lowest speed that doesn’t induce “chatter”, where the tool bounces off the nail. Most Dremels don’t have a setting that’s slow enough to be a problem. but you’ll have to use your judgment on that.
Once he’s used to having you do his nails with the grinder, you might want to try the guillotine cutter (a good quality, sharp one), and see if he’ll trust you with it. I suspect he will. It would be good if you could make the switch. It’s easier and safer.
I should also mention, in case no one else did (apologies if I missed it), that a guillotine cutter, a good sharp one, properly wielded, should never split a dog’s nail. I’ve never used a bad one, and I second Rue DeDay’s recommendation of Miller’s Forge, but I guess if they can design a bad hammer (and they can!), then they can probably design a bad guillotine cutter. I suspect your dog’s bad experience was the result of a dull cutter and/or a poorly (if at all) trained groomer. You might want to warn all your dog owning buddies to avoid that place.
Okay. On preview, I see this is more like $1.50 worth than 2 cents. Can I owe you? 