“Anointed One” is translated as “Messiah” in Hebrew and “Christ” in Greek. Since the New Testament is written in Greek, “Christ” is almost always the term used. However, Jesus and his apostles primarily spoke in Hebrew (or the related language Aramaic) so they would have used the term “Messiah.” You can see this in action in John 1:41 and 4:25, where John explains that the term “Christ” is the same as “Messiah.”
The terms “Messiah” would have probably been the term used until about 50 AD when virtually all Christians were Jews who spoke Hebrew. From about 50 AD to 70 AD, the term “Messiah” would have been used by the Jewish Christians and the term “Christ” would have been used by the Gentile (non-Jewish) Christians. During that time period, the terms would have been synonymous. That is also the time period when virtually all of the New Testament was written in Greek, except for the books by John.
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Peter’s Great Confession happened in an unlikely place. After a lengthy journey, Jesus and his disciples arrived at Caesarea Philippi in the far north of Israel. This was the location of pagan temples. It is possible the entire incident happened within sight of those pagan temples.
It is nearing the end of Jesus’ ministry. Matthew 16:13-20 tells of a key incident. Jesus begins by asking his disciples, “Who do people say the Son of Man is?” In this case, it is clear he is using the term “Son of Man” to refer to himself.
The disciples answer by offering a variety of names. It appears they offer more names than Matthew records. They name John the Baptist, Elijah, and Jeremiah.
Jesus responds by asking the most important question that can be asked. “Who do you say I am?”
Instead of the whole group responding as before, Peter offers his answer. “You are the Messiah, the Son of God.” Jesus replies that Peter’s answer was revealed to him by his Father in heaven. Jesus ends his reply by ordering his disciples not to tell anyone he is the Messiah.