Reading the Church Fathers discussion
Augustine of Hippo: City of God
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Book XVI. From Noah to David
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The following may be redundant or informative, dunno. But to me the definition of a type is a historical thing that also foreshadows a future unfolding of some kind. So for me in a Venn diagram "type" would be a subset of "historical event." And for right now I'm being extremely loose in my use of the word "historical." It includes, let's say, things deemed historical in the Scripture. So it would include Noah.

1. Historical (What really happened, e.g. facts)
2. Significant (What is meaningful, e.g., allegories)
3. Relevant (What is relevant to the Church, e.g. prophecies)


My comment was narrowly about the term "type." The Venn diagram does not directly relate to that term, so far as I can tell. This is what I'm talking about.
"Typology stresses the connection between actual persons, events, places, and institutions of the Old Testament, and their corresponding reality in the New Testament which they foreshadowed. ... Allegory, on the other hand, finds hidden or symbolic meaning in the Old Testament, which is inherent in text and does not depend on a future historical fulfillment ...."
For all I know the West in general and St. Augustine in particular may not use this term. But in the East it's a technical term defined as in the article I linked to.

TYPOLOGY
A method of biblical interpretation by which a person, event, or institution (“type”) in the OT corresponds to another one (“antitype”) in the NT within the framework of salvation history. It was frequently employed by the church fathers and favored by the Reformers. It was rejected by the “enlightened” theologians but reanimated in the wake of the Biblical Theology movement. After losing ground it again is experiencing a renaissance, especially among evangelical circles in North America.
Ninow, F. (2000). Typology. In D. N. Freedman, A. C. Myers, & A. B. Beck (Eds.), Eerdmans Dictionary of the Bible (p. 1341). Grand Rapids, MI: W.B. Eerdmans.

In my first comment, I was using "type" in the sense of model and pattern, which would include allegories, as well as historical entities. In other words, it is a superset of the "technical" type.
The Venn diagram is an attempt to capture Augustine's view of the relation between history and allegory in the Scripture.
In Book XVI, Augustine traces the history of the Two Cities, from Noah to the Patriarchs, and to the time of the Kings.