This particular Booker is one that:
a. took the baptism?
b. beheaded (another) Elizabeth in the deal with the portal
c. felt guilty about said deal and went to Rapture looking for another life, another girl.
This Elizabeth is one that:
a. is PRIME! (check the other thread, I freakin' love the name "Beth Prime")
b. Prime meaning she is the Elizabeth from the main story of Infinite, as she references her events with main Booker often.
c. is vengeful as hell wanted to kill of this version of Dewitt.