Part of the difficulty is in defining terms.
I attempt to use the word "church" (small c) to describe the body of believers, regardless of denomination; and the word "Church" (capital C) to describe a particular denomination.
IMHO its impossible to identify any peculiar denomination as the sole owner of the title "Church", as it will depend on the context of whom you are speaking to. When I use it locally, people will likely understand me to be talking about the Church of Scotland or the Free Church of Scotland; if I were to use it in Italy, likely they would think of RC; in Greece, the Orthodox.
But I would say that, as I'm a Reformer at heart 
Lil:
I think it's a little difficult to be that specific during the first century. While the Apostles were around it was pretty much the case that your denomination was "Christian". Though there were "other teachers" that attempted to form sects (such as the Nicolatians and Judaizers), the sense still seemed to be that you were with us or against us. Every attempt was made to keep everybody together, as seen in the "Circumcision" council of Jerusalem reported on in Acts and James.