I'm just going to describe another way of scribing the score line for cutting your bottle. Please see the rest of Fstedie's excellent guide for instructions on how to candle the break and smooth the cut.
Step 1: The Jig is Up
Lay your wine bottle in the gap and adjust the glass cutter so it rests on the wine bottle. You will be rotating the bottle while pulling the bottle toward the cutter, so it needs to be snug, but as long as you have contact the fit doesn't seem to be all that critical. See the intro for a pic.
I found this nice pro style glass cutter in the gutter - the dinky little .99 cent ones may not want to clamp on as easily, so you may need to make a little cradle for yours to keep it from moving in the clamp.
Note that I got in a hurry taking pics - you'll need to remove the labels from your bottles first. You can cut above the labels for tall glasses and leave the labels intact if you like, but you'll have to be careful with the burning candle part to avoid damaging them.
If you do try some cuts with the label on, you'll probably hit "bumps" as you rotate the bottle since the sharp edge of the Workmate top will want to tear into the paper. Some masking tape wrapped around the bottle in a few places would probably fix this issue.