Messiah is an English version of the Hebrew word that means anointed. Christ is an English version of the Greek work that means anointed or the Anointed One. In the Old Testament, prophets, priests, and kings were installed into their office by being anointed with oil, which seems to symbolize the Holy Spirit.
As the fulfillment of the offices of prophet, priest, and king, Jesus is the Prophet, Priest, and King. He was anointed, not with oil, but with the Holy Spirit and with power (Acts 10:38).
A Hebrew-speaking believer who wanted to refer to the coming Savior as the LORD’s Anointed would call him Messiah (meshiach). A Greek-speaking Christian who wanted to refer to the Savior as God’s Anointed would call him Christ (christos).
The New Testament consistently calls Jesus the Christ even when the speakers are Jews. Some translations retro-translate christos back to mashiah when the speakers are Jews. Our practice is that when the New Testament text has Jesus speak a Hebrew word (as in his frequent use of amen) we follow the text and keep the Hebrew word. If New Testament uses the word christos regardless of whether the speaker is Jew or Gentile, we use Christ. Neither practice gives a wrong meaning, but we prefer to follow the literary choice made by the writers, who were inspired by the Holy Spirit. If they use a Hebrew-based term, we do too, If they use a Greek-based term, we do too.
A special case is John 4:25. The Samaritan woman says, “I know that Messiah is coming.” The text includes the parenthetical comment that the Messiah is also called Christ. Following the lead of the text we include both Messiah and Christ in our translation. When the Greek text uses only christos, we translate Christ.