L. Phillips, Welcome to the forum!
There are different thoughts on whether to run the cutter off the edge of the glass or to stop at the edge. Some are concerned with the chipping & possible damage to the cutter head from the impact of hitting the work surface. I prefer to run the cutter off the edge. To decrease the possible damage to the cutting head & the amount of chipping, lighten the pressure slightly as you go off the edge. Use a softer surface to cut on such as self-healing rubber cutting pads commonly used with crafters or Homasote board. Then in most cases I will run the score starting from the edge where the cutter went off the glass. This can be real helpful on challenging glasses which do not like following scores. The chipping unless you have a heavy hand is generally minimal & by the time the glass is ground, foiled or placed into the came, will not show.
The type of cutter handle shouldn't make any difference. I've used barrel/pencil, modified barrel, Thomas grip, & various styles of pistol grips. The process is the same. You want 4-7 pounds of pressure which will leave a light score line. If you are leaving a white dusty score line, you are pressing too hard. The "zipper" sound you will hear talked about is a nice gauge when scoring clear or smooth cathedral glass but will not always be heard when scoring harder opals. Do not worry if you can not here the zipper as long as you can see the light score line. When starting off, take a piece of paper towel & wipe the scoring fluid of choice off the score line to help you to see the score line as you evaluate it. Sometimes you think there is a score line but it is just a line of oil.
Wish you the best as explore this new adventure!