There is a short but definitive article published in the Journal of Evangelical Theological Society in 2012 What's in a Name? An Examination of the Usage of the term "Hebrew" in the Old Testament written by an OT Professor Dr. Matt Akers which describes the difference between "Hebrew" and "Israel" in the OT, and how both Hebrew and Israel "finally became a racial designation for God's covenant people."
After introducing the pliability of ethnonyms (II), Eber's relationship to the Hebrews (III), and the usage of HABIRU (cognate of Hebrew) in extant ancient literature (IV) the article analyzed Old Testament's usage of "Hebrew" to discern what the author wanted to distinguish from "Israel" (V).
Here's the conclusion (VI):
After analyzing the above evidence, several observations may be made. First,
“Hebrew” and Habiru certainly are cognates. Both words, second, possess nearly
identical shades of meaning. The terms were ethnic designations that over time
began to denote immigrants, warriors, and servants. Third, one of the earliest references to the Habiru hails from Mesopotamia, the region from which God called
Abram the Hebrew. The shared geography cannot be a coincidence.
For these reasons, the populace of the ancient Near East would have regarded
Abraham, Isaac, Jacob, and Joseph as Habiru because of their semi-nomadic
sojourner lifestyle. It does not follow, however, that they would have considered
the Habiru to be Israelites as the lineage of Eber establishes. Simply put, in the
early OT era, “Hebrew” refers to any descendant of Eber, while “Israelite” pertains
only to the branch of Eber’s family that Jacob sired. Only later in OT history did
“Hebrew” finally become a racial designation for God’s covenant people.
Another journal article The Habiru and the Hebrews: From a Social Class to an Ethnic Group published in Volume 7 Issue 3 of the Jewish Bible Quarterly provides a similar conclusion. The article provides additional details on the archaeological discoveries relied on by the JETS article.
Concluding paragraph from the article (emphasis mine):
All the evidence from archaeological discoveries to date seems to point to the
conclusion that, sociologically, the Hebrews were in fact Habiru, although not all Habiru were Hebrews. It could well be that the word עברי (Hebrew) was
originally only a sociological designation, indicating status or class - in which
case the words Hebrew and Habiru are synonymous. The fact that in the later
Books of the Bible and in its usage in post-biblical times, the word Hebrew has
been used as an ethnic designation simply means that the original meaning of the
word has been changed. With the eventual disappearance of the Habiru,
etymological explanations of the term "Hebrew" such as mentioned at the beginning
of this article, were inevitable. In the absence of archaeological evidence until
comparatively recent times, the Pentateuch itself was the oldest record extant
from which an explanation could be sought. And so the term "Hebrew"
ultimately became equivalent to the term "Jew" as in the Book of Jeremiah where
the prophet proclaims:
that every man should let his man-servant, and every man his maidservant
being a Hebrew man or a Hebrew woman. go free; that none
should make bondmen of them, even of a Jew his brother ... " (Jeremiah 34:9)
Nonetheless this cannot detract from the clear indications which exist that the
origins of the Hebrews are as Habiru.