Having once had the fun of removing a jag which was left in my guns barrel after the threads snapped off I have "fixed" all of my remaining jags by installing steel threads.
The problem with the brass threads is brass is pretty weak and if it is subjected to high stresses, it will work harden making it brittle. Weak + brittle = broken threads.
To fix the problem I filed off almost all of the existing brass threads and then center punched a dimple in the center of the remaining thread.
Using the correct size tap drill I then drilled a hole 3/8 to 1/2" deep.
I threaded this hole with the correct tap. (I have both #8-32 and #10-32 range rods. Don't ask me why)
Once threaded, I installed a common steel 5/8 inch long set screw into the jag with about 5/16" sticking out of the face. I used Loc-Tite thread locking compound to secure the set screw in place.
As for which style cleaning jag? I haven't seen a difference between them and I have both styles.