...the true kind of course, is simply anyone who is committed to the fulfillment of the Abrahamic covenant.
Jews were only Jews by belief, and therefore when a gentile was converted to the Jew's belief, he was then a Jew.
These terms tend to confuse because Paul was determined to be all things to all people, and so his use of terminology depended on who he was writing to, and was not intended to establish the proper meaning of specific words.
As far as adoption is concerned, no scriptural assembly from Abraham onward, was dependant upon a bloodline, or genealogy. Every generation of an assembly, considers new members of the assembly to be strangers who are adoped in. In a generation, it is forgotten that they once were strangers.
"Context is always the key."