Birth of the Church?

Today, the Feast of Pentecost, is often described as the "birthday of the Church," but I'm not sure that's the best characterization. It is hard to find an exact beginning point for the church. Certainly Jesus' words "You are Peter, and on this rock I will build my church" are a pivotal moment. One could say that the crucifixion was the beginning--the church being born out of Christ's wounds, signified by the water and blood that flowed from his side. Or you could say that the church was born with the resurrection--just as each baptism is a new birth by being joined to Christ's death and resurrection.

Some problems with describing Pentecost as the church's birthday are:
1. It would suggest that there was no church (sacred assembly) before Pentecost
2. It would suggest that the Holy Spirit was never given before Pentecost
3. In the analogy of the Holy Spirit descending upon Jesus at his baptism, it would suggest that Jesus was "born again" in some fashion.

I think we could understand Pentecost best as more of a commissioning and equipping of the church than a birth. The preface for this feast sums it up well:

". . . through Jesus Christ our Lord; according to whose true promise the Holy Ghost came down on this day from heaven, lighting upon the disciples, to teach them and to lead them into all truth; uniting peoples of many tongues in the confession of one faith, and giving to thy Church the power to serve thee as a royal priesthood, and to preach the Gospel to all nations."