Thanks for your responses. I solved the mystery of the "mid-cycle" posting of miles.
I went through the online process to change my payment closing day from the 22nd to the 2nd. (The process seemed to stall midway, and I didn't get a confirmation that the change had gone through, but apparently it did). The change resulted in a statement being issued immediately, so the miles for my first purchase then posted right away. In other words, the miles didn't actually post mid-cycle.
I'm glad the change went through; now I know I can make the minimum spend on at least the Visa card by the next closing date. The Amex card is presenting more of a challenge. But I have always excelled at spending money, so I think I will be able to figure out how to do it.
Did you change the closing date to the 2nd before or after 10/2? Just wondering if you change the date to a day that has already passed during the month if it triggers an automatic statement right away. For my next churn I want to have my US air miles post ASAP otherwise I'll have to pay to have old miles not expire and this could be a good way to do it.