There are a couple of things that can make a difference in your cutting. I've found that using rulers from different manufacturers can cause problems. If you have rulers from different manufacturers, check them out to see if their lines line up with each other. Also, mats are not always equal in distance between lines when compared with rulers. I only use the lines on the mats to make sure that my material is sitting straight on the mat. I always use my ruler for measuring. How you hold your rotary cutter can make a difference. My quilting buddy tilts her blade in towards the ruler. I have my blade up next to the ruler (not tilted). That can make a difference. (Consequently, we don't cut strips for each other because of how we hold our cutter.)
The thickness of your needle and the thickness of your thread can make a difference when you press your seams one way or another. Make sure you are consistent with which machine you use and what thread you use.
You asked about where to put the ruler on the material...on the line or to one side...I honestly think it doesn't make any difference as long as you do it the same way each time you cut. I tend to put it right in the middle of the line but they may just be me.