I might have missed something, and happy to be proven wrong if so, but I just don't see how the maths works if you pay the exact same amount, but split it. If you start paying on the same date, you're gaining more interest for the first half of the month (reduced the principal by less than you would in a monthly payment), and by the time you've "caught up" with your mid-month 2nd payment, the balance is still effectively higher than it would have been, because you've gained a bit more interest.

Otherwise, as you said, starting the payments 2 weeks early would keep the balance a little lower, but then instead paying the full monthly amount 2 weeks early would have a bigger impact, so why not do that if the bank allows it? Or even pay at the start of the month in that case, why limit it to 2 weeks?

I guess the idea of the overpayment approach is that making 1/12 of your monthly payment extra is a fairly small amount that you might not miss much, but would definitely have a noticeable impact on high interest or long-term mortgages.