[core] Fix theming bugs

[apedroferreira]

Attempting to fix remaining theming bugs:

• Docs and app use same storage key for theme mode
• Console error (mismatching style tag) when app loads from server in dark mode if defaultMode is not set to dark (even though there seems to be no flash with wrong theme, I don't think this happened before MUI v6 either)

[apedroferreira]

I'm pretty sure the console error mentioned above is because the playgrounds aren't using v6 of Material UI, so it should eventually be solved once mui/material-ui#43001 is solved.
We can merge the fix for the storage key "bug" and this should be good!

[apedroferreira]

@Janpot just adding a few more small changes in a few minutes (added now, just some more props so I assume it should be good to merge)