as you can see there’s no icon on Safari new tab page when Twitter is in favorites.
This is because an « apple-touch-icon.png » is needed, like a favicon.
It is also used on iOS when people add a website on their Home Screen.
Sometimes it appears on iOS, but never on OS X Yosemite.
But anyway this is a quite old icon.
I think this problem can be easily solved and I’d be pleased to see it on my screens