Adding a new outlet

In another post I presented a project I am working on in my home. I should have gone about the question in a different way so am going to start over with a new post. I am not an electrician, but have done other wiring projects in my home without any problems.

I have a 40 amp circuit in my shop for a radial arm saw. The 8 gauge cable for this circuit runs from the circuit breaker box to the shop and ends with a dryer type outlet (sorry I don't know the proper technical term for that). I want to add another outlet of the same type between the existing outlet and the breaker panel for another machine. I plan to cut the existing cable and added a box (4" x 4" x 2") at each of those cut ends (2 boxes total). At each of those boxes then "splice" (again that is probably the wrong term – sorry) using appropriate sized wire nuts a new length of the 8 gauge cable to go to the new outlet. Those 2 new cables are then connected together at the new outlet. The junctions at the 2 boxes are being made with wire nuts that are appropriately sized for 8 gauge cable (I have tested making these connections with some scrap pieces of the 8 gauge and they are good and tight).

Am I doing anything horribly wrong here? Here is a crude drawing of what I want to do. Thanks

Here is the original post:

I just realized one other bit of information I left out. For the new outlet that I am adding I was planing to use the same 4" x 4" x 2 1/8" boxes that I used at the splices. Is that ok? Does that box need to be bigger?

Here is a couple of pictures of the existing outlet, from what Harper said (few comments down) I think that maybe this is not a proper outlet?? (I did not install this one, it was done by an electrician).