OK I looked at your tool again. The clamp as designed by Des Burke the originator of the Diamond Tool Holder, has a very interesting clamp.
The Clamp is designed to clamp either round or square stock. The clamp is not quite square and uses Vee style stamps on three sides. I have yet to ever have a tool bit slip. even on the hardest of materials. I have never pushed it to make extreme DOC just to remove metal fast. I have attempt to copy some of the other designs and found that by making short cuts on the design, I have found in every case the bit did have a tendency to slip. This is a subject of much debate on some forums where some would rather build their tools rather than buy the actual DTH. In my case I found the cost of the tool saved me a lot of frustration, In actual use and the sharpening and maintaining the tool bits. I have two sets of both the Left and Right tool holders.
I am convinced, at least in my head, that a split clamp using one or more screws to provide the clamping force to retain the tool bit in place.
If the tool is aligned to cut a stationary Work Piece in the same manner as the DTH does on the lathe.
A person has to view the cutting process from a static aspect. In my opinion.