It turns out the solution is quite simple. You just need to visit the website in private mode to make Safari refresh/reload everything including the favicon for bookmark. And then when you go back to non-private mode, everything is refreshed. Phew.
1. Quit Safari.
2. In the Finder, select Go > Go To Folder.
3. Enter~/Library/Safari/Favicon Cache/and press Return.
4. Use Select All (Command-A) to select the database and cached items.
5. Drag these to the trash or press Command-Delete.
6. Empty the trash.
7. Launch Safari.