I'm not sure what answer you think I've given to the belief in G-d thing here.

That was my mistake. I missed the word 'not' - which was a fairly crucial word to miss - and now I'm unable to edit it. I was trying to say 'you've not actually given an answer'. My point being that given the question 'Do you believe in god?', your attempt to answer it didn't actually answer it.

What is it that they all have in common which allows us to apply the term "Jew" to them all? I'm arguing that it's that heritage of Torah & Mitzvoth, even if part or all of that heritage is rejected at an individual level.

Sorry, I find that somewhat ridiculous. That's like claiming that the heritage of eating meat allows us to apply the word 'meat eater' to everyone even if they don't eat meat. The fact that their ancestors might have done and, even, they themselves might have at some point in the past does not allow anyone to apply the term to them now. If someone followed the Mitzvoth in the past - or never did but had parents who did - you don't get to just call them a Jew. It's silly.

A Jew is someone whose mom is Jewish. That's not just some opinion; that's the legal definition.

Whose legal definition? Wait, you mean the Jewish legal definition. Which is, essentially, opinion. It's certainly not the law of the land (at least, not in the land I am). Again, if you make your own definition and then apply it, no-one can argue with it. It's another tautology. You're also essentially saying that anyone ever born in the female line of someone who's ever been identified as a Jew is then Jewish ad infinitum. Absolute rubbish. So when a Jewish mother births a daughter, who moves abroad and cuts ties with the family, who births a daughter of her own, who becomes a Muslim and moves to Tunisia who births a son who is brought up in a Muslim family and has never even heard of Judaism... that son is a Jew.

No.

But the world won the argument by wearing me down through repetition.

Call me a bluff old sentimentalist but I prefer to resolve arguments through reason and critical thinking. I can't imagine any point in my life where someone standing in front of me and repeating "You're a Jew." will ever convince me that I'm a Jew. As far as arguments go, that one's shit.