@billz5557,
All this talk about wanting a "pristine" daughter who is "intact", and your family thinking that you now see her as "dirty" or "impure" really sounds very sexual to me--as in "pristine" and "intact"=virginal, and now, because of her ear piercings, she is somehow defiled and "impure".
That's a rather extreme way to view a third ear piercing. This seems to have symbolic significance for you that is way beyond the reality of the situation. And because of the possible symbolic sexual significance you are attributing to things like this, your family may well be unable to to understand your thinking about it and it is the reason they have labeled you a "prude".
Try to view this situation in terms of basic reality. Ear piercings are ear piercings, and nothing more than that. They do not reflect on your daughter's reputation, her social class, or how sexually active she is. They are a form of adornment you might or might not care for, simply in terms of visual appeal, but they are nothing more than that. Several ear piercings is a style that has been around for quite some time, it is hardly radical. The style apparently appeals to your daughter. Your wife is apparently accepting of that.
There was no way your wife or daughter could have discussed the piercings with you in advance. Your mind has been closed on this issue for a long time. You admit that, if they had told you what they planned to do, you would have talked your daughter out of it. That means you were not prepared to listen to your daughter at all. What she wants, and her reasons for wanting it, are not as important to you as your being able to control her, to get your own way. You are cutting off communication between yourself and your wife and your daughter. And very likely, you are doing this on other subjects beyond ear piercing.
Your daughter is not a child any more. She is a 16 year old young woman with opinions and ideas of her own. It is entirely normal for her to express those things in the way she dresses and adorns herself, and it is entirely normal that you might not always approve of her choices. These are matters you can discuss with your wife, and you have to be willing to listen to your wife's views on these things as well. Your daughter has to be allowed a certain leeway to experiment, because that is part of growing up and it is a part of her separation from her parents. She will want to adapt styles and trends of her peer group, and that is perfectly normal. As long as she does not harm herself, or create unwanted problems for herself, there is nothing wrong with trying to fit in, because that is important at her age.
Your daughter has a long, long way to go before she becomes a physician and before she is looking for marriage, and a lot is going to go on in her life between now and then, and you are really going to be able to control very little of it. Your major job as a parent who can shape your child is already done. You can continue to try to keep her safe, you can continue to offer her guidance, but you cannot continue to try to mold her into an image that conforms with what you want her to be. Becoming an adult is a process of self-discovery and, as much as possible, you have to encourage that without imposing your own wishes and needs on her. When you start being overly critical, or overly judgmental, about her choices you will start shutting down communication with her and you will damage your relationship with her.
I suggest you sit down and have a long talk with your wife. Ask her where she thinks you are being wrong with your attitudes concerning your daughter, and really listen to what she has to say. Then try to see the situation through your wife's eyes. Your wife was once a 16 year old girl, and she may have a perspective on things that you really need to understand. I suspect that you might be at odds with her over more than just this ear piercing business. You may well be unreasonable because you listen with a closed mind. You have to be willing to acknowledge that your wife's point of view may be as valid as your own, and you have to be willing to at least try to change how you see things if that will help you to understand her, and your daughter, better. Open, honest communication in your family is more important than a third ear piercing. If you are, in fact, a "prude", face that. If you are, in fact, too controlling, face that. If your rigid attitudes are creating problems within the family, try to change that.
It's not easy being the father of a teenage girl, and you're likely to hit a lot more bumps in the road beyond this latest ear piercing. Just hope that none of them are any more serious than this one. And, if you really want to surprise your daughter, offer to buy her a new earring for that third piercing. It might cause her to view you in a whole new light.