Nope, canon disagrees with you. Once Jean said yes they became one and then it became an extension of her natural abilities. They aren't separate right now. That's why Jean is in the white hot room as white phoenix of the crown. Jean judges universes and if they are determined to be dead ends she wipes them out. The Phoenix is a conscious force but Jean is it's prime master. Storm would have to pull an omniversal feat like meghan to be in Jean's league.