I thought that to be eligible to play for a country, a player needed to be eligible for immediate citizenship.
Under Israeli law, all Jews are eligible for immediate citizenship.
To he considered Jewish, a player needs to trace his lineage through the matriarch (a child born to a Jewish father and non-Jewish mother is not Jewish [unless the mother converted in a very specific manner {at least under Israeli religious law}]; a child born to a non-Jewish father and Jewish mother is automatically Jewish). (Name, baseball skills, location have nothing to do with it.)
Additionally, a player himself who is not born Jewish, can convert through a very strict process. Israeli religious law differs from, let's say, American Reform Jewish laws, in that Israeli conversion law follows strict Orthodox procedures, while American Jews have a whole menu of procedures to choose - some less "strict" than others. American conversions are NOT recognized in Israel, unless done through the Orthodox method.
Now, during its history, Israel has absorbed immigrants from many countries (e.g., Soviet Union, Ethiopia) where records of the ancestral line were missing or not reliable. At various times, other methods of proof were accepted (e.g., an entire community knowing Jewish law and following Jewish customs was sufficient (immigrants from Africa) - but only following much public comment, uproar, and argument.) This has lead to some controversy as immigrants fleeing oppression in some areas insisted they were actually Jews in order to settle (there was one example about 50 years ago where a group of African-Americans from Chicago insisted they were Jewish and refused to leave Israel; after years of battles, that group was allowed to settle as Jews.)
FWIW, in any given year there are about 60 - 100 professional Jewish players scattered through the landscape of affiliated ball.