I have been a member of the church of Christ for 35 years, and my family has been a part of the church for at least 5 generations. I have never heard anybody use those terms! There are people all over the world that follow God's word. They may or may not have a sign nailed on the wall where they worship that says "Church of Christ, but they are Christians according to God's commands.
Old Testament history and prophecy is complicated, and I have not studied it enough in recent years, but the only remnant I've ever seen discussed in the Bible is the remnant of Israel. Could I be a part of that remnant if I'm not Jewish?
"They may or may not have a sign nailed on the wall where they worship that says "Church of Christ, but they are Christians according to God's commands."
You are saying what I posted just using different words.