What does ''halachically Jewish'' mean?

Answer 1

Halachically Jewish means Jewish according to Jewish law. In Jewish law, a child born to a Jewish mother or someone who has converted to Judaism is considered a Jew; one does not have to reaffirm their Jewishness or practice any of the laws of the Torah to be Jewish. According to Reform Judaism, a person is a Jew if they were born to either a Jewish mother or a Jewish father. Also, Reform Judaism stresses the importance of being raised Jewish; if a child is born to a Jewish father and was not raised Jewish then the child is not considered Jewish. According to Orthodox Judaism, the father's religion and whether the person practices is immaterial. No affirmation or upbringing is needed, as long as the mother was Jewish.

Answer 2

It means the definition of who is a Jew according to Jewish law.

Jewish tradition states that a Jew is any person whose mother was Jewish (Talmud, Kidushin 68b), or any person who has gone through a proper conversion to Judaism (Talmud, Yevamot 47a).

Answer 3

Halachically Jewish means Jewish according to Jewish law. Jewish tradition states that a Jew is any person whose mother was Jewish (Talmud, Kidushin 68b), or any person who has gone through a proper conversion to Judaism (Talmud, Yevamot 47a).
It may be noted that despite remaining technically Jewish, one who does not practice the religion is not viewed as a Jew in traditional Orthodox Torah-sources, unless they repent ("Our nation is a nation only through its Torah" [Saadia Gaon in his Emunot veDei'ot; and many similar sources]).

Answer 4

"Halakhically" means "done in accordance with Torah-law".