The following is my understanding of a consistent Reformed understanding of Christian marriage, though it likely applies to others outside of the Reformed camp.
A marriage is a public, legally binding civil covenant recognized by God that binds together a male and female couple into a family that is consummated by sexual relations. It was created by God to be a model of his ultimate relationship between Christ and the Church (Ephesians 5).
This means that there will be some leeway in the precise marriage traditions and ceremony from culture to culture, though there are clearly biblical constraints on marriage and universal moral requirements of marriages that transcend culture. The Bible recognizes marriages between non-Christians as real marriages as a kind of common grace institution. It also recognizes that there are immoral marriages and non-ideal marriages that are possible.
This is why contemporary gay marriage debates should not be argued on the grounds that marriage is a religious institution. Though Christians might have a religious basis for scrupulously avoiding theft, the public, civil recognition of property ownership is not, therefore, a merely religious institution.
For a parallel, consider ordinary contracts and covenants in the Bible. We see all kinds of traditions at play for how people would ratify them: cutting animals (Genesis 15), exchanging sandals (Ruth 4), grabbing the thigh (Genesis 24:9), and so on. Yet they are all doing essentially the same thing but with different symbolism, occasions, and tradition.
Similarly, marriages can be culturally nuanced and yet bindingly so. In a culture where it was normal to deal with a woman's father and then immediately consummate your marriage without ceremony, this was a legitimate and proper way to begin a marriage (Isaac, Jacob, etc.) whereas it wouldn't be in a context where marriages are not ratified this way. For example, for Christian marriages in America, the marriage is traditionally considered "ratified" when the officiating minister declares the couple husband and wife after the exchange of vows. If this couple privately met a week earlier and rationalized having premarital sex on the grounds that they were getting married anyway and biblical couples were "married in the eyes of God" this way, they would be fornicating because in the "eyes of God" they should have officiated their marriage in the way that public covenants are normally ratified.
We could imagine a culture where business contracts were officiated by exchanging sandals. If you helped yourself to your side of the deal without following this sandal ritual this would be stealing in the eyes of God, not because sandals are necessary to prevent stealing but because contractual commitments matter to God.
This understanding should reconcile all of the Biblical examples of marriage while also addressing contemporary traditions. Marriage covenants needed to be "official," however the society generally officiates such matters.
For the issue @davidlol raised in a comment:
John 4 18 in which Jesus told the Samaritan lady at Jacob's well that
she had had five husbands but the man she now had was not her husband
may be relevant here, although I don't know how. What did Jesus mean
that he wasn't her husband?
This is now easily answered. She was living with a man for whom she had not followed the appropriate culturally recognized method for officiating a marriage.