I self banned myself from SF.

[Evoni]

Is the Bellagio Bellini another name for the signature Bellagio Cocktail?

I don't think so, just my weird way of phasing it. Bellagio by the way serves their Bellini's both frozen and on the rocks, I prefer frozen but both are good. The Bellini is peach puree or nectar with a sparking wine like Prosecco. Some bars will of course make it with different flavored fruit purees.