There are two goals to aim for when distributing your weight:
- You should be in balance, for obvious reasons; and
- The combined centre of mass of your body and the bar should only move down and up during the squat, not forwards or backward, as forwards or backwards movement would cause lost efficiency.
Keeping your weight over the centres of your feet achieves both these things. In terms of balance, you are as far as you can be from either tipping forwards (having your centre of mass move forward beyond your toes) or tipping backwards (having it tip beyond your heels). In terms of avoiding forward-backward movement of your centre of mass, the mid-foot is an easy point of reference for where you should feel the pressure in your feet in order to maintain the transverse plane position of your centre of mass.
The tripod idea, in which the weight is distributed between the heel, the medial ball (first metatarsal head) and the lateral ball (fifth metatarsal head) of the foot, is a more recent one. It aims to add the requirement that the weight be distributed evenly from side to side of the foot, reminding the lifter not to push with the inner or outer edges of the foot. Your intuition that this idea should not result in a more forwards centre of gravity is correct. If you want to describe your weight distribution like this, it would be with 25% of your weight on each heel, and 12.5% on each medial ball and each lateral ball.
As for whether the lateral arch of the foot should be bearing weight, that will really depend on the shape of your foot and your shoe insole. I don't think it's something you should try to focus on or control.
Finally, do you have some mind cues (for the way up and down) to help
distribute the weight on the three points in a proper way?
For forwards-backwards balance, just focus on keeping the pressure in your feet over the centres of the feet.
For lateral balance, try standing up (without a barbell) and supinating your feet so that the medial arches rise up off the ground and all the weight is on your lateral arches. Then do the opposite, pronating your feet so that your lateral arches rise and all the weight is between your big toes and heels. Now flatten your feet again. Remember the feeling of these three positions, and try to avoid the first two during the lift.
If you find that you have a problem with excessive pronation or supination during a squat, you may need a specific cue to counter that, e.g. "weight on your big toes!" if your problem is excessive supination.
