If you watch the meeting of all the Dons, you will see Tattaglia keep looking to Barzini to take the lead...and you see Barzini literally take over the meeting.
Another tipoff: Vito opens the meeting by thanking Barzini for "
helping me to arrange this meeting [emphasis added]." But Vito's little helper doesn't act like a mere helpmate. He sits at the head of the table; takes the lead in indicting Vito ("His are not the actions of a friend...he must let us draw the water from the well..."); declares an agreement made as soon as Vito says he's willing to do whatever's necessary for peace; brokers the embrace of Tattaglia and Vito-- and (nice touch!) is the only one of the Dons who smokes a cigarette instead of a DiNobli, a sure sign of an up-and-coming modern Don. Meanwhile Vito was at his most humble. He made Barzini show his hand. Too bad he didn't figure out Barzini beforehand.