I started John Van Seters' In Search of History.
I'll start this post with Van Seters' "criteria by which to identify history writing in ancient Israel", which he enumerates on pages 4-5.
The first one is as follows: "History writing is a specific form of tradition in its own right. Any explanation of the genre as merely the accidental accumulation of traditional material is inadequate." This criterion is significant because it forms the backdrop for Van Seters' critique of the traditio-historical method, which (in some tellings) focused on individual units of tradition, as if they were oral at some point and accumulated elements before they were finally written down by a passive redactor. Van Seters does not view Herodotus or the historians of the Hebrew Bible as mere compilers or redactors of fairly coherent stories, but rather as active authors, who drew from sources to write their history.
The second-third criteria deal with the purposes of history: it gives "significance for past events" and tries to account for the state of the present, often by looking for moral causes---"who is responsible for a certain state of affairs?" The fourth criterion is that history is "national or corporate in character", meaning that it deals with a nation, not just a king. And the fifth criterion is that history "is part of a literary tradition and plays a significant role in the corporate tradition of the people."
Some of these criteria pop up in Van Seters' comparison of the Hebrew Bible's historiography with that of Herodotus, or other Greeks. On pages 39-40, Van Seters states:
"Like Herodotus, the Old Testament exhibits a dominant concern with the issue of divine retribution for unlawful acts as a fundamental principle of historical causality. [W]hat is significant is that historiography, whether in its various forms in the Old Testament or in Herodotus, deals extensively with law as an important element in understanding the actions of men and nations and the consequences of those actions...While the so-called Deuteronomistic historian and the Chronicler are concerned with this issue only on the national level, and so are more restrictive in scope than Herodotus, the Yahwist and the Priestly Writer do indicate the universal dimensions of the problem."
This overlaps with Van Seters' criterion of moral causation. I'm not sure how the "national or corporate in character" criterion comes into play here, though. To be a history, does a work have to deal with one nation primarily---the nation in which it is produced---or can it have a more universal focus?
On page 51, Van Seters states:
"...in terms of the scope of subject matter and the themes treated, nothing in the literature of the Near East before the fourth century so closely resembles the biblical histories as the Greek prose histories. Both deal with recent events, such as the Persian Wars or the Exile, and their causes through successive periods of the past. Both reconstruct the distant past through the technique of genealogy development with anecdotal or folkloristic digressions. The combination of 'official' sources, such as chronicles, with oral tradition, and of poetic fragments with prose narration in a multigenre product, is not evident in any other body of preserved literature in the Near East in this period."
This overlaps with the criterion that history tries to account for the state of the present. But Van Seters also highlights here a significant point of his book: that the Hebrew Bible can be compared with Greek historiography, for the "earliest Israelite histories" date to the sixth century B.C.E. (page 8). Van Seters does not believe that the earliest Israelite history was composed in the tenth century and "anticipated the emergence of the Greek historians by five hundred years" (page 8). Rather, Van Seters holds that Hebrew historiography occurred during the time of Greek histories, which included the sixth century B.C.E.
Van Seters dismisses the likelihood that "there was much direct cultural contact between the Greeks and the Hebrews before the fourth century B.C." (pages 53-54), so how does he explain the similarities between Greek and Hebrew historiography? Van Seters' solution is that Phoenicia served "as a bridge between Israel and the Aegean as well as a center for the dissemination of culture in both directions" (page 54).