“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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According to Mark 14:61-62, at the trial of Jesus, Caiphas asked him, “Are you the Messiah, the Son of the Blessed One?” The “Blessed One” is a euphemism for God, since the Jews often refrained from using his name. The question of Caiphas is actually two different questions, “Are you the Messiah? Are you the Son of God?”
The first question had at least two purposes. It was against Roman regulations for someone to claim to be the Messiah. That was rebellious talk punishable by death. Also, the people believed the Messiah was to live forever, so crucifixion would prove to them that Jesus was not the Messiah.
The second question was a ploy to get Jesus to commit blasphemy. By admitting he was the Son of God and quoting the verses from Daniel, Jesus was claiming to be God himself. This was blasphemy, a crime punishable by death according to the Jewish law. However, the Jews did not have the power to enact the sentence.