You should definitely have styptic powder powder there to stop the bleeding if you cut too far.
Also, in the vetmed link (good info, BTW), they show a cut that's kind of diagonal or even vertical on the nail. This is from a side perspective as if the dog were standing.
I was taught to put the scissors or clippers along the bottom (underside) of the toenail and just cut off the part of the nail that protrudes through the clipper. That way you are less likely to cut into the quick. You're just cutting off the "hook" that curves below the bottom of the nail.
Don't know if I explained that well enough, but if you look at these photos from the vetmed link, you can see the yellow line is kind of at a diagonal angle. I wish I had access to a photo program here -- I'd draw another colored line extending the bottom flat portion of the nail out through the front. That's the area you should be cutting (to avoid the thin end of the quick, which is hard to see).
[img]http://www.vetmed.wsu.edu/ClientED/images/dog_nails/label.jpg[/img]
In this picture you can see how they're cutting almost perpendicular to the dog's nail.
[img]http://www.vetmed.wsu.edu/ClientED/images/dog_nails/whitenail.JPG[/img]
And finall, when it's cut, you can see how it gets close to the top thinning (and hard to see) portion of the quick.
[img]http://www.vetmed.wsu.edu/ClientED/images/dog_nails/correct_light.JPG[/img]
I would try to hold the clipper more like the following picture, placing it up along the underside of the nail.
[img]http://www.vetmed.wsu.edu/ClientED/images/cat_nails/clipper2b.jpg[/img]
I hope the pictures make this description at least somewhat understandable. This method seems to be an easy and safe method for vets to teach and pet owners to use. It may not be as aggressive of a cut, but for those us us that don't do this to ten dogs a day, well maybe we should be a bit more conservative. Our breeder on the other hand has been doing this for decades and can make a much more aggressive cut for a shorter nail without hurting the dog.
Even so, occassionally even professionals cut a little too close. Your pup will forgive you. A sore nail is probably better than a repeated pulled muscle from sliding around on hardwood floors.
EDITED for typo. Ran out of time, but I'm sure there are others....