to be technical you never played as a true big daddy in the first game. you are only wearing the suit of one and had some of the pheromones of one, the only thing that happened to you that is done to a true big daddy is the voice box alteration you go thru, other than that you're still a human
I found out something interesting. In Bioshock 1, when you find Dr Suchong's body...pinned to his desk by a Big Daddy Drill. One thought can go into your mind after you listen to his audio recording showing his demise at the hands of a Big Daddy. The Bouncer theory was debunked because we've never seen a Bouncer without its drill. Rosie Big Daddy's have no drill. The Logical conclusion is that when Dr Suchong insulted and smacked a Little Sister, he acidentally angered the said Big Daddy which was revealled to be an Alpha Series. (The Big Daddy that you are in the boots of.)
This is known that Dr Sucho wa indeed killed by an Alpha Series.