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.