Are we missing an important element here? The issue is extremely complex, requiring all sorts of nuances.
It's not complex at all, unless you want to muddy water and manufacture problems that don't exist.
All Christians are to some extent sinners; either in morals or maybe even in ideas (I am an OEarther, and perhaps others would not call that a sin; still they think it an error)
All Christians are sinners in the sense that were all born sinners and separated from God and now we have been legally declared right before God. Sin has not yet been eradicated from our lives and all Christians still sin from time to time in a sincere attempt to serve God. But there is a difference between stumbling while trying to serve God and practicing habitual sin as a lifestyle.
Therefore, I think no one would doubt that each adjective in the following list must be removed at some point (this life, or upon death, or upon the resurrection)
But my guess is few would deny the "Christian" label on account of the "sin" attached to it. That is, their vices do not invalidate their belief in Christ. If it did, we are all doomed and there can be no such thing as grace.
All of those are contradictions in terms. Can Christians stumble in those areas? Yes and they do. But there is a difference between committing a sin and living in sin as a lifestyle, as I mentioned above. If someone can live habitually in sin and enjoy it, they need to reexamne the authenticity of their profession of faith. A person who lives in pornography, a person who habitually steals, who regularly and inpenitently sows discord through gossip, a person with no self-control over their most base appetites needs to get right with the Lord.
I would also point out that living in sin doesn't ususaly encompass just one type of behavior. A person living in the sin pornography ususally has other things going on as well, that they are living in that are just as sinful. Sin takes over your entire life, not just one part of it.
Does a person practicing a homosexual lifestyle (let's omit all factors like promiscuous sex and keep it to marriage) still have a relationship with Christ if he or she claims faith in Christ? Does Christ acknowledge him or her as a disciple, yet in major error in this regard?
No, because the first thing Jesus requires of a person who wishes to become His disciple is that they repent of their sin. Jesus expects them to make the quality decision to turn from their sin and after that Jesus expects that they will bear the fruit of repentence. They need to first of all realize that homosexuality is a sin, and that their sin separates them from God and they need turn from it. And if they have repented from the heart and truly turn from their sin to serve God it will show up in their lives.
Jesus is not going to accept them as His disciples while tolerating their homosexual lifestyle. That is not how it works. There is no pardon from sin without first their being genuine repentence ake place. Jesus doesn't tolerate sin. If He did, then there would have been no reason for Him to die to save us from sin, in the first place.
....is the homosexual Christian still a Christian; does his or her belief in the appropriateness of homosexuality completely invalidate his or her faith in Christ?
There is no such thing as a "homosexual" Christian, it doesn't matter what sin you tack on to the title of "Christian," the result is the same. It is a contradiction to think that person can live in habitual, unrepentant sin and still claim to be an authentic follower of Jesus. One reason the Church is so messed up today is that we have allowed people to remain comfortable in their sin. Preachers have abandoned the Gospel to preach on things like, managing finances, how to find happiness, how to make your dreams come true, and they no longer preach on sin, on the coming judgment. They never talk about man's accountability before a holy God and that they don't tell people what it really means and how much it really costs to serve Jesus. Preachers preach what sells, not what is true.
The Bible says this:
No one born of God makes a practice of sinning, for God's seed abides in him, and he cannot keep on sinning because he has been born of God.1Jn 3:9
The person who thinks they can live anyway they want and that Jesus will accept them based on a verbal profession of faith without the need to repent or be transformed is a sorely deceived person. Those who practice sin are not Christians, bottom line.