To actually launch and then play Living Skyrim, follow these steps:
Open Mod Organizer 2 via the ModOrganizer.exe in your Living Skyrim folder.
In the top-right of Mod Organizer 2, select "Living Skyrim 5 [SKSE]" in the dropdown menu.
Click the large Run button to launch the modlist.
The dropdown and Run button. (LS4 Placeholder image)
If you receive a prompt to update the game's registry path, click "Yes". If you don't receive this prompt, you've likely already done this and can carry on.
Registry popup, click Yes when prompted. (LS4 Placeholder image)
If you'd like, you can also use the "Shortcut" button below the Run button to create a desktop shortcut for launching the list.