Want to know how hex bit shank works and the history behind it's design
I'm going to throw out some opinions here that I don't have hard evidence to back up in form of citations.

The Wikipedia article you liked kind of mentions this, but the reason hex shank (6 sides) is a thing is most likely because drills normally have chucks with 3 jaws.  A shank with 6 sides is a natural result of that.

The extendable bits you show with the small notches are only good in very low force situations.  I have a driver with a bit like that and it never gets used because when I really need to press on a screw, the bit slips into the handle.  The small notches are designed so that a metal ring can tighten around the shank in the grooves to hold the bit.  The grooves are small, so it can only support so much weight.

The larger round groove in the hex shank is made to hold the bit from slipping in or out and the deepness of the groove means it can support much more weight.  The circular profile allows locking to be done with a thick ball bearing rather than a thin metal ring.  Ball bearing locking mechanisms are found in many well made padlocks because of their simplicity and ability to resist very strong forces.



Ball bearing locking mechanism from LockLab

The 1/4" diameter is based on the typical torque these tools are used at.  Cordless drills are the main users of hex shank, and then driving screws is the main end goal.  Screws have a maximum torque they can accept before breaking, and the drill obviously has a maximum force it can apply.  It's arguable that todays technology can over-torque a 1/4" shank, but in the past a cordless drill just wasn't that strong.  Anything thicker would be overkill for the common task at hand.