I usually just make my lace out of bridle leather, that way you can cut whatever width you want. Then I bevel the top edges and dye the edge. Then I split it down to the weight I want, I think it's easier to dye the edge while it is full thickness. Then I lube up the lace with some type of saddle soap, I use glycerine or Tattersal's. Lace it up, go back and tighten the lace.
On English reins, to finish, you just skive the ends of your lace and put 5 or 6 stitches in to hold the ends of the lace.
If I was doing those reins pictured, I would wrap the ends around the rein and splice and stitch them together in a loop so it would look like the beginning of the lacing and then work the slack back.
Hope that makes some sense,
Kevin
PS You don't have to worry about burnishing the lace, that happens naturally as you pull it through the holes.