Christ is by definition not a name, it is a title.
It appears to be used as both in Scripture. When accompanied by the definitive article (often appearing in our English translations as "that" or "the") it appears to be a title. But it is not always used that way. For example, the Jewish leadership and others used it as His name - see passages such as Matthew 26:67-68: "Then spat they in his face, and buffeted him, and others smote him with rods, Saying, 'Prophesy to us, Christ, Who is he that smote thee?'" In the Greek there is no definite article before "Christ" here which occurs when it is being used as a title - "the Christ" - like in a previous verse in this passage, 63). This (using Christ as a name, not a title) occurs several more times in Matthew's gospel as well as the other gospels (for example, Pilate does not say "Jesus who is called the Christ" in Matt. 27 - Pilate twice says "Jesus who is called Christ."
That's why in the NT the phrase "the name of X" almost invariably includes Jesus, while frequently not mentioning Christ.
But there are also many times where the opposite occurs - Christ (with and without the definite article) is frequently used without mentioning the name Jesus.
in the psalms it sometimes refers specifically to the Davidic king and not to Christ, except in a secondary sense
I was surprised to see you say this. Although referring to a different term in Matthew 22, Christ clearly teaches that the
primary sense of many of the Psalms was actually referring to Him: "While the Pharisees were gathered together, Jesus asked them, Saying, "What think ye of Christ? whose son is he?" They said unto him, "David’s." He said unto them, "How then doth David [in Psalm 110] in spirit, call him Lord, saying, 'The Lord said to my Lord, Sit at my right hand, till I make thine enemies thy footstool?' If then David call him Lord, how is he his son?" And none could answer him a word, neither durst any from that day forth ask him any more questions." (Matthew 22.41-46). Note what Christ said - David was speaking/writing "in spirit." How much of the primary meaning of the Psalms was missed by unbelief or not having the Spirit in the same fulness that we do? Isn't this what Christ was promising in John 14.17: "the Spirit of truth, whom the world cannot receive, because it seeth him not, neither knoweth him: but ye know him: for he dwelleth with you, and shall be in you."? Do you also think Psalm 2.2 is referring to Jesus Christ in a secondary sense? Considering vv. 7 and 12 that follow, I think it is clear the Divine Messiah is clearly being referred to specifically.
Outside the psalter the title can be used even more broadly, for example of Cyrus
Agreed. Just as the name Jesus/Joshua/Josiah are broadly used.
Again, I just don't understand how anyone can sing the Psalms without thinking about Jesus Christ the whole time.