Wrong diagnostic o' mine.
Besides doing what's stated above, I had also re-enabled all Samsung bloatware that I had previously disabled.
I've just investigated more thoroughly and found out that disabling app "S Planner" stops Galaxy S3 from syncing with Facebook.
It looks like this app doubles as a bridge between app "Calendar Storage" and Facebook. Without it, sync works (no error) but it doesn't store anything.
Minutes ago I had all Samsung bloatware disabled again, including S Planner, and Facebook calendar wouldn't sync. After enabling S Planner... bingo.
It's common sense that installing app "Google Calendar" calls for disabling S Planner. Just like a moth going into a candle's flame.
Other ways of having your calendar NOT synced: deleting the Facebook app "Samsung Galaxy S III" in Facebook (via web) or changing your password in Facebook.
So to make sure your Facebook calendar gets synced in Android:
1. Enter Android settings.
2. Turn off Auto-Sync.
3. Remove your Facebook account entry.
4. Make sure app "S Planner" is enabled.
5. (optional) Maybe it's a good idea to have app "Samsung Push Service" enabled too. Just a hunch.
6. Turn on Auto-Sync.
7. Exit settings and run the Facebook app.
8. Log in and grant all of app "Samsung Galaxy S III"'s requests.