Are you being baptized and confirmed on the same occasion? I ask this because, while people do not usually use their confirmation name in everyday life, they often do use their baptismal names, although, depending where you live, you may need to change your name legally for this to have legal effect (in some countries a baptism certificate alone may be sufficient). If you want to be known as Rutherford Vincent you could consider using Vincent as an additional baptismal name as well.
To answer your question, I suspect that an element of personal choice comes into it. I’ve never heard of this name badge situation you will have, so it is presumably just a custom at your particular parish. Perhaps ask your priest or someone else responsible for RCIA. What I can tell you is that my confirmation name was John the Evangelist and when I was actually confirmed the priest just said “John”, so don’t be surprised if the priest or bishop just says “Vincent”. If you want to make it clear on your name badge which Vincent you chose I can’t see that there would be a problem with that. Best to ask in your parish though, as this is not something everyone will be familiar with.