I agree to use the method that works for you. I was cutting fabric like you until the instructor said it was backwards too.
She explained a couple of reasons to cut with the ruler over the fabric. Lines on the mat can be off a titch. Repeated cutting on the mats can leave grooves in the mat. If the rotary cutter slips, you are cutting into the strip you need. If you cut into the yardage, you just re-cut the straight edge. These weren't big issues in my view but I did mange to change methods.
Now I cut with the ruler over the fabric. I use the grid lines on the mat to square up the edges.
This is one of my favorite cutting tutorials. It makes sense that we really aren't backwards at all. She mentions that the horizontal line of the ruler doesn't matter at 2:00 minute mark. I do check the horizontal line.
https://www.youtube.com/watch?v=XfvKE0Y6npk