the ketubah is the contractual marriage. It is a legally binding contract. Once it is signed, you are married.
The ayrusin is the betrothal. Back in the day, it wasn’t even performed in conjunction with the nisuin – they were seperate, and the ring was basically a symbol of the commitment (although that’s putting it a little lightly, it was more like a guarantee of marriage, you were off the market etc.). When the nisuin finally came around, you could finally move out of your parents house and be together. And it could be months after the betrothal. Obviously its different in these modern times and we do both together, but the ring does not mean you’re married (unlike in many non-Jewish weddings where ring plus vows means you’re married).
As far as yichud, while it used to be the time when you consumate the marriage, typically you don’t do that anymore. And whether you do or you dont have sex, you’re still married one way or the other. Its an important part of the conclusion of the ceremony, but it doesn’t nullify the marriage if the expected doesn’t happen (and its generally not expected anymore).
So there are many different symbolic and legal gestures we make in the process of the ceremony, but there arent really 3 ‘ways’ to get married, you definitely can’t pick and choose (legally I mean…practically, anyone can do whatever they want as long as all parties, including the rabbi, agree), and I guess there is some flexibility as far as the order of things goes, but not very much (again, legally speaking…and again, everyone can just do whatever makes them feel good).
I used to be frum and am obviously not involved in that community anymore, so I’m certainly never one to tell people what they should and shouldn’t do with their ketubah (I mean, non-Jews have ketubot now, so…). If someone wants to sign it mid-ceremony, as long as they and their rabbi are cool with that, then go for it. However, its always good to know what the technicalities are so you can make an informed decision. The ketubah is the contractual marriage. If you’re cool going in without having the contract signed, then by all means, go for it.