The yellow arrow points to a leather cord tied similar to a poacher's knot to have an end loop.
The red arrow are ends of an elastic string which passes through a hole (one for each end) in the leather.
I need to tie the elastic string ends in such a way to secure both the leather cord, and on itself so that when the elastic is pulled, it will not stretch. Also it should be hard but not impossible to untie.
As a climber I have to disagree, figure 8 knots are the best, especially when double back.
It's the one knot I trust my life to on a weekly basis and I have rarely encountered a situation where they weren't part of the solution.
Look, friend, if you can't trust 15 year-old me, who acceptably tied FOUR different knots that one time to check off the box in the handbook, or 17.83333 year-old me who managed the building of not one but TWO picnic tables for a local elementary school because my Dad was being a pain about finishing the final Eagle Scout project, then who can you trust?!?!?!?!
If you want to be able to trim off the ends, I suggest a double fisherman's instead (aka the double overhand stopper) with both strands (hold them together while you tie and treat them as a single strand).
With either knot, after you trim, you can melt the ends of the elastic cord. It will "mushroom" a bit, which will also prevent the tips from creeping into the knot and coming loose.