I carved some initials into the badge but we didn't film that part. It's essentially a bit of low relief carving, the basics of which you can see here and in the the lettering of the child's room sign here.
The carving on the cap is light, ie shallow, and the background punched to give a textured effect. Angle the walls of the relief letters out a little to catch the light. You will of course need small tools. Having said all that, the world in your lobster: you can carve anything you like on the badge! I would suggest you keep any design simple. Bear in mind that we don't have colour for contrast and we depend on light and shadow to make the badge readable.
There are videos on preparing and using beeswax here. Seal the underside of the cap with a matt (acrylic) varnish.