I tested two X10 RF remotes {HR12A and UM19A} with both a CM15A and an RR501.
Observed power line signals with a Smarthome 1132CU.
CM15A with both RF devices sends the command twice on the power lines and one time from the on screen buttons.
RR501 only sends the commands once for each RF command.
I believe the remotes send the command twice to ensure a better operations. It seems like the CM15A also sends a power line command twice for RF commands received.
It could be you have marginal power line signals to the WS13As and between the first and second set of commands from the palm pad they work. While the single command from the screen does not.
I don't have any data on timers sending commands more than one time.
Update:
The way the CM15A handles the RF signals seems to be age related. My old ones do not transmit the power line command twice. The later ones do.
Date Code: 04J41 Controller IC P10792E does not.
04J44 Controller IC P10792F does not.
05C10;05D18 and 08B09 Controller IC P10792M does send the power line command twice.