As I understand it, the theory is the first shot is a well-constructed expanding round to do maximum broadside damage through the heart/lungs and the second shot is about penetration on either a butt-shot (aka, Texas heart shot) or a charging buff through the skull. That theory though is decades old and the recent trend I've seen has been to use well-constructed softs or TSX's flat across the board and let the PH handle the solids if needed.
I haven't shot Cape Buff yet but when I do (hopefully next year) I'll be packing 300gr. Partitions, A-Frames or TSX's 5-down in my .375. What I do know is that a 286 gr. Partition at 2500 fps punched right through North American bison broadsides without bothering to stop or ask for directions.
Correct that theory is very old and also stems from the time when premium grade expanding bullets were not available.
With a proper expanding bullet you can take broadside, quartering or full frontal shots in an appropriate caliber.
Lets take the average first time buffalo hunter who wants to back up on buffalo after his first shot. Most times you will be hunting in a herd. After the first shot the herd will thunder off. If you are very lucky the one shot at will be going away directly away from you(mostly it will be to the side) now you cannot take a shot if you are backing with solids, unless it is going directly away from you. With premium softs you can as long as you are sure you can hit the right buffalo.
Where exactly should one then aim if using a solid in order to reach the vitals? Anywhere and hope for the best? From the rear aiming for the vitals on a running buffalo is no easy task.
With a premium expander you have three visible targets you can shoot at, both hip bones and the spine, just above the root of the tail. This will drop or anchor the buffalo allowing for a finishing shot or worse case slow the buffalo down considerable.
Again here the importance of correctly and accurately placing the first bullet is the most important.
You do not need a solid to sort out a charging buffalo. Premium expander again is a better choice.
When he charges the head is up with the nostrils flared. Any shot in line with the eyes all the way down to where the neck vertebrae dip at the lowest point is going to drop him. Often the aiming point is the nose. I prefer just below that.
Only time you will need to shoot for the brain is at the very last moment when he dips his head to engage you with his horns and even then you do not need a solid to punch through to the brain. You could also shoot for the center neck shot above the horns when he dips his head.
This is assuming a Premium grade expanding bullet in an appropriate caliber.
A last point to consider, when using expanding bullets never ever take a quartering away shot from the left hand side as the first shot on a buffalo.
All the best, have fun.