From reading this, I'm not sure if you're interpreting it as casting on a separate set of 10 stitches not joined to the 10 you've already knit. You don't. You do what you thought when you said -
....use my working yarn from the ten stitches I just knit
I'm talking about the right strap here. You've knitted 10, then you use that working yarn and a new piece of yarn to cast on 10. After the cast-on that new piece of yarn is cut to about 8" and left dangling. Later it will be used to make the button loop.
For the left strap, yes, you cast on a new set of 10 stitches with your yarn and a second piece of yarn which is cut to about 8" and again left dangling. You continue on to knit the 10 sts that are on the needle for the left side of the bootee.
Now I may well have just reiterated what has been said, but it wasn't clear to me exactly what was meant..... so sorry if it was already clear to you.
It is a little confusing - I've done these booties and got rather confused too, and I've read that others have too, so you're not alone.