The right click move to another page is the supported route rather than a workaround, so you are not missing a cleaner in-menu method. Dragging between tabs in the same group is the part that should be working, and that is the bit worth catching in the video basj asked for.
On the under the hood question, yes. The pins live in the registry and where depends on which menu style you are running.
Windows 11 and App styles: HKEY_CURRENT_USER\Software\Stardock\Start8\Start8.ini\GroupContents11\$$APPS$$
Windows 10 style: Start8.ini\GroupContents\$PINNEDDEF$ under the same Stardock\Start8 branch.
Windows 7 and Modern are the odd ones out. Those pins are not registry entries at all, they are shortcut files in %appdata%\Microsoft\Internet Explorer\Quick Launch\User Pinned\StartMenu.
Two practical notes if you go in there. Refresh Regedit with F5 after each add or remove, otherwise you are looking at a stale view and you will misread whether your edit actually took. And take a settings backup first from the Settings tab in the Start11 config, which writes a .S11Backup file. Rebuilding a menu layout by hand is a miserable evening.
What I am not going to do is tell you which value maps a pin to page 2 specifically. I would be guessing at the layout in there, and a wrong guess in the registry costs you more than it saves.