(I hope this is the right way to reply)
Good question. No, of course, it does bump up the principal. How about this thought experiment: for continual compounding, what if the payment is not made continuously? I only mean this rhetorically. The fact is, the payment period and the compounding period do not need to coincide at all; the compounding period is a convention. I hope I express this properly, and I do not do this on a day to day basis, but my understanding is:
 The compounding per se is on a six month basis, but can be calculated at any time.
 So, if you are late three days (and no grace period, etc), the amount payable is added to principal for this period of time. If you ask the bank (and their systems are up to date), you can determine new principal, which is old principal plus payment amount plus accumulated interest to that day.
 If you pay on the third day, you will have interest payable on the original principal plus the delayed payment for three days at the six-month compound rate (expressed as a fraction of days / six months).
 In fact, this is basically identical to interest on a given sum for x% annual interest for y days. You would pay interest of principal times x% times y/365. It just doesn't get compounded for the interim days (for example, on a daily basis).
 Why six months, etc? I have no idea. That was the convention a long time ago and it has stuck, and I think at one point "they" chose to require banks etc to all quote and calculate on the same basis, rather like an APR.
 The point being any rate/compounding period can be chosen, the calculations are just extra work often dictated by convention/regulation. Hope this is clear.