One of the classes taught at Westminster Seminary California is titled "Ancient Church." It covers the history of the church after New Testament times until the Medieval period. I have sometimes joked that it is really misnamed. We should call it the "Somewhat Old Church" because I teach about the really ancient church in Old Testament classes like Pentateuch and Historical Books.
Now some might disagree, arguing that we don't find the word church in the Old Testament, and they would be correct but only for the English. The Greek word that is translated as church is ekklēsia and is used in the New Testament for a specific gathering of Christians, or for all believers together. The same word, ekklēsia, is used in the Greek translation of the Old Testament, the Septuagint, to translate the Hebrew term qāhāl, "assembly." It is used for Israel as they gathered together to appear before God or as a general reference to all of Israel. Another Greek word used to translate qāhāl is sunagōgē. It is also used to translate another word for Israel: 'ēdâ, "congregation." In the New Testament sunagōgē is most often used of a synagogue or its members, but is used for a gathering of Christians in James 2:2. The terms used in the New Testament for the church are not new ones but are rooted in the Old Testament. Thus, in at least some sense, Israel really was the Old Testament church-assembly-congregation!
And yet, we could say that the church stretches back further, even before Israel. The church includes Abraham who was summoned out of the nations (Gen 12:1), the line of Seth who called upon the name of the Lord (Gen 4:26), and Adam and Eve who embraced God's first gospel message (Gen 3:15). The church could even include Adam and Eve as created in perfection to dwell in God's presence, but we will not pursue that now.
The Heidelberg Catechism says much the same thing in Q&A 54: "Q. What do you believe concerning 'the holy catholic church'? A. I believe that the Son of God, through his Spirit and Word, out of the entire human race, from the beginning of the world to its end, gathers, protects, and preserves for himself a community chosen for eternal life and united in true faith. And of this community I am and always will be a living member." As we think about what the Heidelberg is teaching, it is helpful to remember that we can speak about the church in two ways. The Heidelberg is speaking of what we usually call the invisible church, those "chosen for eternal life" - the elect. But as we think of the church in the Old Testament, we are usually thinking about the visible church, the various gatherings of the covenant community in different times and places. The Bible makes clear that these gatherings can include not only the elect but also those who profess with their mouth but not with their heart.
These various gatherings of God's covenant community, the visible church, look quite different throughout history. Most of what is described before Sinai is very family-centered while with Israel we find a theocratic nation. Why? In summary, God has formed his covenant community in different ways throughout Redemptive History by his covenants and laws to point us to Christ and his consummate kingdom. Thus, there is some wisdom in using the designation of the church for God's covenant community after the coming of Christ to distinguish its unique place in Redemptive History. So, maybe we don't need to rename our classes at Westminster after all!