Me summarizing t the conclusionh: Jesus called Himself an Israelite, Judaean was a term primarily used by non Israelites to describe them. Israelite was the preferred in group term. Jesus thought of His mission as to the children of Israel. Paul and other NT data, and external data confirm the preferred usage of Israelite over Judaean. In addition, the Greek Ioudaioi is best translated as Judaean, not Jew for historical reasons. Using Jew and Christian ( “Christ lackey” or “Christ partisan” similar to Herodian “Herod lackey” or “Herod partisan” creates false mental associations in the minds of readers, and ideas of “conversion” etc. Paul didn’t convert, “religion” didn’t exist as a separate institution from politics and family/tribal groups, Jesus and all early followers were Israelites entirely. The Jesus movement was in the beginning and thought of itself as an Israelite renewal movememt. Judaism should not be used in translations, but rather “a Judaean way of life” as Judaean was used in ancient times as an outsider designation of the people in that land, not some religious designation. In addition to all of this, Jews are a post-mishnaic group, and Christians in the modern sense began centuries later. Calling people in the Bible Jews also causes false readings of the conflicts of the early Jesus movements and the other Israelites, making openings for anachronistic anti-Semitic readings. Israel is also preferable for grandeur and antiquity reasons, since it is used in the Torah to designate the group Jesus is from.
Also Christian was not used as a self designation in those times.
He provides evidence for all of these conclusions in spades, so it is best to refer to Jesus and His early followers as what they thought of themselves: Israelites. It is best not to use anachronisms.