I have been cutting glass since I could reach over the glass bench
Family business.
The green handled cheap glass/tile cutters are pretty nasty and gouge out a cut. Yes they may get the job done but a good quality oil filled single diamond wheel cutter is far better. But, they are about £15-£30 for a decent one.
Like using most tools its experience and practice rather than the all about the tool its self.
Even pressure and let the cutter do the work. You are not trying to cut the glass in half with the cutter but rather create a fine even score and weakness in the surface that can be vented and hopefully split where you want it to. To vent a cut you tap from the opposite side of the surface you have cut. I would use the back of the cutter. Has to be something metal and take your time.
Always makes me laugh when you see the cat burglars cut a circle in one side of the window and pull it out. Complete tosh. You need to have access to he back of the window which kinda defeats the object
A bottle is going to be a bit of a pig as you want a continuous score and pressure on something that's round. Stopping and starting is where you are going to trip up so johns idea about mounting the cutter and turning the bottle sounds like a plan to me.
If you struggle to find a good cutter let me know and will pop in and see how much they are at my mums place.