To help one understand this a little better one needs to know how that Ecclesiastical position
- defines the people of God throughout time
- places the covenants (both biblical and theological) in relationship to each other, and
- especially the relationship between the the church with respect to Israel.
In Classical Covenant Theology (held by some church fathers - 100-500 A.D.) there is only one people of God for all time (both the Old and the New Testament), so they see the church =ing Israel. In the Old Testament the people of God or the OT church was limited nationally (to the people of Israel) and physically (eventually to the promised land of Canaan). There is only one overarching covenant in the Bible that has different administrations and renewals expressed in the Biblical covenants. This view emphasizes the continuity (or unity) of scripture between the OT and the NT. This affects their end times view by eleminating any view that would necessarily separate the people of God, since there is only one new man (Romans 11). This makes their end time options normally Amillennialism (the majority position of Christians throughout church history) or Postmillennialism with a few holding to Historic Premillennialism (the Premillennialism of the church fathers rather than Dispensational Premillennialism). The chart is meant to show these options and relationships. They tend to flatten out the Bible equalizing the Old Testament and the New Testament. They believe in covenant children and that there is a place in the covenant of Grace for unbelieving children of believers until they denounce Christ and his gospel. They would still say that only true believers are saved. The problem for me is that they don't distinguish between Physical Israel (most of whom died and went to hell) and Spiritual Israel (believers in the Gospel and in the coming messiah) sharply enough. They tend to spiritualize the Old Testament in deference to the New.
In Classical Dispensational Theology (first created around the 1830s) there are 2 people of God with different promises and different covenants give to each, so they see the Church <> Israel. Thus there is no OT Church since the church is a completely different entity that the people of Israel. There is no overarching covenant in the Bible, so each covenant is self contained and ends when another begins (hence the term "dispensation"). In the extreme side you find people who believe that there are different ways of salvation for the Jew and Gentile. this view emphasizes the discontinuity (or disunity) of scripture between the covenants and between the Old Testament and New Testament. This makes their end time view Pretribulational and Premillennial, because they necessarily created and needed a "rapture" before the Millennial Kingdom to keep the Church and Israel separate - hence they call this age the "church age" because of its a Gentile purposes and focus, and they call the next age the "millennial age" being a physical age belonging wholly to Jewish purposes and focus. They tend to prioritize the Old Testament over the New and use more of a physical hermeneutic rather than a literal hermeneutic which spiritualizes the New Testament in deference to the Old Testament. The chart is meant to show these options and relationships.
The chart is my attempt to show the necessary relationships between ones Ecclesiology and Eschatology. For me I fall somewhere in between these extremes, since I believe that the promise That God gave to Abraham and his seed and that seed is not a physical people but Christ (Galatians 3:16) - hence those found in Christ from both the OT and NT (understanding the progressive nature of Biblical revelation and that previous revelation must be viewed through the lens of the fuller expression of later revelation). I don't see the Church in the OT (while believing that there has always been only one people of God by faith alone), but see the Church as a New Covenant People that now includes both Jew and Gentile. I see both unity and disunity or continuity and discontinuity in Scripture rather than one or the other (I guess this makes me more and more a Baptist.) I also tend to be a Historical Premillennialist but with Amillennial sympathies. As one can see I am still working out my salvation with fear and trembling.