On Expatriates, on my userpage it says I have earned the inquisitive badge,
but on the inquisitive badge page it says it has been awarded 0 times
The badge pages are cached. Those pages reflect information from every user on the site, so recalculating them every time somebody earns a badge would be impractical. The notification on your profile only affects you, so you get it immediately.
You might have noticed that even when you get that notification on your profile, your badge list (down at the bottom of the Activity tab) doesn't always reflect the new badge right away, either. The badges are updated separately from the notification.