Hi Bryan
I'm not sure if I get 100% what you're saying, if I'm interpreting your question correct but here goes, hopefully this helps you out:
In that last screenshot, if you select a line (click on the gray block in front of the line) you can use the upper menu "Goto - Details" to show the determination of that specific partner profile. 
On the next screen, you can then check which Access Sequence is assigned to this specific partner function.
Access Sequences are build in another spot (transaction SPRO - define access sequence) and such an access sequence defines what the system looks for, in which sequence. However, it's an "OR" story so if you an access sequence with three sequences (sequence 10, 20 and 30). The system looks into sequence 10, if he finds a business partner that matches, he inserts it and does not check 20 and 30.
So if you want to do something more intelligent / complex than what is available in the basic functionality / customizing, you'll soon end up using the BADI capabilities to have more complex logic to assign business partners so you'll have some coding to do (or to get done) if you want to use that.
There might also be alternative methods to reach the same end-result but I cannot say that really without investigating the actual situation. I hope this already helps out to some extent so you can progress further.
You can find information on SCN on Business Partner Determination, Access Sequence etc to get a better understanding how the functionality works. I have resorted to try and fail, try again and in the end succeed ways of working with the functionality because it's not well documented from my perspective and not always clear what an access sequence does exactly when you pick one from the list etc.
Reference:
BAdIs in Solution Manager ChaRM 7.1
Specific BADI in question: COM_PARTNER_BADI
Best regards
Tom