Using getDatabasePath doesn't work without extra work on older API versions.
That location doesn't exist, and it seems like we'd need to add a bit of code
to check for its presence, create it, etc., before we can rely on that location.

All of our other code which stores files just uses `filesDir`, and I don't see
a reason not to put `places.sqlite` into that location as well. There isn't much
value in that file living in a "databases" folder.