The clue is the renaming, as well as the other distinctions given Peter alone: the keys, the charge to feed the sheep, and the promise to pray so that Peter converts his brethren. This means that Jesus meant his Church to be the effort of Peter in some personal and unique way. Your interpretation: "This is a big rock, and you are a small rock" does not fit the broader context.
I disagree. Peter was never seen in Scripture as the Leader of the entire Church. Shortly afterward, Jesus is rebuking Peter and saying "Get thee behind me Satan!" Paul rebukes him as well. Regardless of the rebukes, Peter is nevertheless NEVER seen as THE head of Christ's church on earth. A leader, yes. THE leader, no. And since Scripture was written into the 90s, and the book of Acts covers the ACTS OF THE APOSTLES, that's a pretty significant omission.
The Rock was Peter's bold confession about Jesus. It's the truth that HE, Jesus, is the Messiah. Upon that truth he will build His church. Peter was called a rock because of his boldness, steadfastness, and unyielding spirit to confess something that could have gotten him stoned. Peter knew the truth. If fact, he knew who the TRUTH was. And Jesus was pleased. He did not make him (Peter) the governor over all the church. In fact, in the broader context, Jesus gave the Keys to the church itself (as is seen just two short chapters later), not Peter exclusively.