Faucet-mount filter thread/size incompatibility
It turned out that the faucet end and the filter were both 55/64-27. I was able to confirm this when 13/16 and 15/16 both did not fit. The lesson is that if it screws on, it's probably the same size. The various faucet sizes are different enough that they won't be able to thread onto each other.
But why was the filter not fitting well despite the threading compatibility? I studied my filter closely. You thread it onto a faucet by turning a metal ring. It's my belief that the ring needs some sort of flange on the faucet to brace against in order to seal correctly. As you can see in the original pictures, th current faucet head consists only of threads.
The solution I took was to use two adapters. First, a 55/64-27 F to 15/16-24 F adapter onto the faucet. Then, a 15/16-24 M to 55/64-27 M adapter onto the first adapter. Then finally the filter. In the picture below, the M/M adapter is not really visible, but it connects the filter to the first adapter. Prior to buying the 55/64-27 F to 15/16-24 F adapter, I thought that I might be able to thread the adapter in reverse (the larger side facing up) so that it could provide a flange on the faucet that I could twist the filter's ring against, but that did not work; not enough room.