I did NOT cut it to fit. That is the way the strap was made from the factory. It's a 20mm strap and notched so the strap edge at the lug extends over in front of the lug tip. Actually it's a Timex replacement strap I purchased a last year. I have used it on another watch for a while, and then replaced it with another strap. So this one went back into the "Strap Closet".
Regarding the pushers, yes, they are for controlling the digital display; setting time; setting alarm; setting the date; and the chrono feature.
Now, I have had to trip a 20mm strap to fit on one of my vintage Timex's which has a 19mm lug width. What I use is an Xacto hobby knife. It has a "pencil like" handle and a triangular blade that is VERY shart. Just move SLOW and EASY when trimming so the cut is clean and neat and you don't cut too much.
Hope this helps. Thanks for looking!