Am I in a domestic partnership or de facto marriage?

I moved in with my girlfriend 6 months ago. We had been together for 2.5 years before that. I am an Australian citizen, she is a US citizen. We live in Brooklyn, NY. We are both on the lease for our current apartment. Am I in a "domestic partnership" or de factor marriage with her? Will I be soon if we cross some threshold of time spend together? I want to understand what the legal implications may be if we break up in, say, a year or 5 years from now without ever getting married. Will that be the same as if we had never moved in together, or will there be other rights that she has to my assets because of our relationship status in the eyes of the law that I may not know about.