Use file system for settings and fix data home path

It's possible that using browser storage makes more sense with settings
sync, so we might want to revisit this once/if we get settings sync
working. As it currently is though, browser storage just causes jank.

The path was also missing a `User` at the end so I added that. This
might affect the Vim extension which would have been writing to the
wrong path previously but I don't believe it should affect anything
else since they would have been writing to browser storage.

- Fixes #2208
- Fixes #2231
- Fixes #2279
- Fixes #2274