What's the actual difference between greater vs metro cities?

https://lemmy.world/post/15411189

What's the actual difference between greater vs metro cities? - Lemmy.World

I hear some cities call the larger area as metro (Vancouver) and other cities call it greater (Toronto). Is there a functional difference? Generally speaking, is one more urbanized than the other?

I always took it like Manhattan vs all of NYC. Manhattan and maybe Brooklyn is the metro part, and all 4 boroughs are the greater city.
Funny, I see it the opposite way.