The picture shows my favorite method for putting electrical boxes in cordwood walls.
I prefer deep metal boxes because they have room for a GFCI and room to stuff the wiring. Some of the shallow boxes are not deep enough for GFCI's, as I discovered the hard way.
For a while I used a router to cut the logs for the electrical box. My 1/4" router could just barely cut deep enough for a shallow box -- but I have since switched to the deep boxes, as discussed above.
I tried using a chainsaw to do the job, and it can be done, but it's easy to screw up, and generally pretty awkward.
I think the ideal tool would be a 1/2" plunge router, guided by a template. However, a 1/2" router is still on my wish list.
So this is my best method to date. Take a log about the same size as the box, and use a table saw to cut the outline of the box. Then use the table saw to slice and dice the rest of the wood to be removed, and use a chisel to clean up the rest. A couple of drywall screws hold the box in place.
I run romex inside the walls, but your local code may require something more substantial. Personally, I am comfortable with the romex.
Embedding the wiring inside the wall takes some planning, and it slows down the cordwood, but once the cordwood wall is done, all that is left is to intall the receptacles and cover plates.