If you are controlling a fan then you should be sure to use a switch designed for fans. These switches are able to handle the greater current draw of the fan motor, as opposed to the more modest current draws of lighting.
If you have a single switch controlling the fan and light, then you will need something that can separate the fan and light control at the fan AND you will need to maintain power to the fan so that the connected control is always powered. Recently I discovered a Wink enabled Hampton Bay wireless fan and light control. Unfortunately it uses Zigbee and not Z-wave, but this should still work in my situation where I have a single switch controlling all power to the fan and light. I will need to re-wire the setup so that the switch cannot turn off power to the ceiling fan and light, but I have not yet found a replacement wireless wall control to fill the outlet from the original switch.
Lutron Pico Remotes work very well in this use-case. You need a Lutron Smart Bridge (preferably get the Pro) and some work to get this set up on ST. I use Pico remotes with Hubitat which has built-in support for the Lutron Smart Bridge Pro, and it exposes Pico remotes like any other button controller. (I am not sure how to get Pico remotes to work in ST, but I am guessing there are plenty of threads on the topic.)
I have the exact same fan wiring as you. I added a Hampton Bay Zigbee Fan Controller, and put the Pico remote in the spot where the old wall switch was (I hard wired the fan power.) With one 5-button Pico remote, I can controller the Fan Speed, Fan Light (including dimming), and a pair of table lamps with Sengled Color Element Plus bulbs. It works remarkably well.
And can you show us a screenshot of the automation which is working the way you want from the app?
If I understand you correctly, it may not be possible to do what you want if the switch directly controls power to the lights. In that case the physical switch on the wall doesn’t go through any of the rules you have set up in SmartThings, it just reports afterwards what it did.
The reason the automation would still work is because the automation goes through the hub first and the rules get applied before the signal is sent to the switch.
If you want rules to be applied to the physical switch on the wall before power is sent to the lights, then you have to change the wiring so that the switch is not directly controlling the power but rather just sending a message to the hub.
The problem with that method is that it means if your home automation system is not working properly, then the wall switch won’t work. This is why most smart switches have a default where as long as they are actually wired in line, they continue to work just like dumb switches when physically manipulated.
All of that is why the brand and model makes a difference.