It's an interesting contrast to a place like Israel.
I've been told (by Israelis) that there is no "civil marriage" in Israel. That is, there are no government marriage licenses, no justices of the peace to marry people, etc. Marriage is solely the province of the religions and their traditions. Israel recognizes marriages performed in other countries, of course, but that gives the odd result that people in Israel who don't want to be married in a temple (or church, mosque, handfasting oak grove...) go elsewhere (Cyprus, for example) to have a civil ceremony and a short vacation.
Indeed, and not only that but so much law depends on the status of marriage. Even if there were no civil marriage, plenty of regulations treat married and unmarried couples differently (income tax, for example)
Consider C. S. Lewis' idea that calling someone a Christian based on whether or not you believe they are following Christian doctrine is not a reasonable definition of Christian. C. S. Lewis says that one must distinguish between Christians who follow doctrine (good Christians) and those who don't (bad Christians) but not call the latter non-Christians.
"Now if once we allow people to start spiritualising and refining, or as they might say 'deepening', the sense of the word Christian, it too will speedily become a useless word. In the first place, Christians themselves will never be able to apply it to anyone. It is not for us to say who, in the deepest sense, is or is not close to the spirit of Christ. We do not see into men's hearts. We cannot judge, and are indeed forbidden to judge. It would be wicked arrogance for us to say that any man is, or is not, a Christian in this refined sense. And obviously a word which we can never apply is not going to be a very useful word. As for the unbelievers, they will no doubt cheerfully use the word in the refined sense. It will become in their mouths simply a term of praise. In calling anyone a Christian they, will mean that they think him 'a good man' But that way of using the word will be no enrichment of the language, for we already have the word good. Meanwhile, the word Christian will have been spoiled for any really useful purpose it might have served.
We must therefore stick to the original, obvious meaning. The name Christians was first given at Antioch (Acts 11:26) to 'the disciples', to those who accepted the teaching of the apostles. There is no question of its being restricted to those who profited by that teaching as much as they should have. There is no question of its being extended to those who in some refined, spiritual, inward fashion were 'far closer to the spirit of Christ' than the less satisfactory of the disciples The point is not a theological or moral one. It is only a question of using words so that we can all understand what is being said. When a man who accepts the Christian doctrine lives unworthily of it, it is much clearer to say he is a bad Christian than to say he is not a Christian."
That's C. S. Lewis' point, I think. If someone says they're a Christian but then breaks into your house, sits in your kitchen eating your food and tries to steal your TV we'd say that they're a Christian and a bad person, but if they, to use your example, reject the foundational doctrine of salvation through Christ's death on the cross, then we'd say they were something other than Christians. Deists, perhaps.