benmychree said:
The core is a "bell center drill"; instead of the 60 degree angle, the bell center drill has a curved cutting edge roughly bell shaped; that being said,
I have no idea if they are still available, I have only ever see/owned one. I doubt that a small offset such as a Morse taper or similar taper would have much effect on center wear or part geometry.
We always called this type of center "Protected Center Drill"
The 60 degree edge was protected from being damaged by the shoulder left when the center drill was drilled about an 1/8" deep from the end of the shaft