Most of the books I've ever read on it (Mafia Dynasty, Five Families, The Most Evil Mobsters in History,) list Vincent as the formal boss. Like the man it's based on, the book "The Last Testament of Lucky Luciano" is full of different "facts" on the subject. In one spot the book speaks of Vincent as the boss but in another place it talks of the Brooklyn outfit run by the Mangano brothers.

For what it's worth a 1963 Senate organized crime chart does show the deceased Anastasia as successor to both the Mangano brothers.

My take is that Vincent was the boss and when he wasn't around Phillip spoke for him, even though Albert was the underboss and had more power than his supposed bosses.

"When the snows fall and the white winds blow, the lone wolf dies but the pack survives."
Winter is Coming

Now this is the Law of the Jungle—as old and as true as the sky; And the wolf that shall keep it may prosper, but the wolf that shall break it must die.
As the creeper that girdles the tree-trunk, the Law runneth forward and back; For the strength of the Pack is the Wolf, and the strength of the Wolf is the Pack.