You're doing it "right". The bulk of your fabric is under or behind (to the left) of the ruler. Only the strip you are cutting is "exposed". All it does is anyway is keep you from whacking up more of your fabric if you you slip with your rotary cutter. Whatever works.

Ditto on everything for me.

If you end up with the square you expect and all your fingers intact then it doesn't matter how you get there.

Which ever works for you is the best way...

Either way works. Just be sure you cut all of your strips using either the lines on the ruler, or the lines on the mat. They are not always the same.

I was using my mat lines to measure my pieces and then I read on the Board not to do that. I went to my cutting board and measured the lines with my metal ruler and found that the lines on my mat are not all one inch. From now on I will not use my mat for measuring my strips.

I use the lines on my ruler, even for the first cut. I lay all the fabric to the right, lay the ruler on so that the first strip would be completely under the ruler at the correct distance, a little wider than needed. That first strip is not usually completely straight on the left side. Cut that off and flip it over; put the ruler back on that piece and cut off the extra from the edge.
That way the large piece of fabric is always on the right and doesn't have to be flipped over. You just flip the first strip to cut off crooked edges. I stand the ruler up on edge and bump it up to the cut strip to be sure the strip is straight, or line it up to a line on the ruler.

If it works for you, don't worry about it!!!

I cut the way you do, but to square it first make sure you edges are straight not just the ends.

This was how I was taught and have always done it this way...doesn't mean this way is the RIGHT or ONLY way!!