Edge is only one part of the new smartthings architecture, and is specifically for devices which communicate directly with a smartthings/Aeotec hub. The vast majority of those are either Zigbee or Z wave, although there are a few Wi-Fi devices as well.
There are still dozens, maybe hundreds, of other devices which communicate with smartthings over the Internet, what we called “cloud-connected devices.“ you can use those with a smartthings account whether you have a smartthings/Aeotec hub or not.
Lutron Caseta falls into the second group. The Lutron devices talk to their own Lutron SmartBridge, the SmartBridge talks to the Lutron cloud, the Lutron cloud talks to the smartthings cloud. So edge is not involved in that at all.
The Philips hue system is unusual in that you get a choice of two different integrations. It can either use a cloud to cloud integration very similar to the Lutron integration. The Philips hue devices talk to the Philips hue hub, the Philips hue hub talks to the Phillips cloud, the Phillips cloud talks to the smartthings cloud. Again, you don’t have to have a smartthings/Aeotec hub to use this integration.
Or if you want to, and you do have a smartthings/Aeotec hub, you can use a LAN integration which allows the smartthings hub and Phillips hub to communicate to each other over the local Network without going through clouds. In that case, an edge driver will be used.
Both Philips hue integrations work well.
Z wave devices like the two that you mentioned will require an edge driver. Of these, I think the Zooz is a little more popular, but you can find discussions of both in the forum.
Are you already using the smartthings app? I couldn’t tell from your post. The reason I ask is that there are hue scenes, Lutron scenes, and smartthings scenes, and they are not quite the same thing. So I wasn’t sure exactly which scenes you wanted to change between, as that could affect the device selection.
The official smartthings/hue integrations, for example, do not allow you to control Hue scenes from smartthings.