To be fair, I carefully read the Chochmos Adam. He absolutely emphasizes intent, namely , that the issur is when someone says “because he (the goy o.m.)dresses like this, I will dress like this, etc…. The examples he gives are plentiful, and all of them are directly related to a desire to be like the goy, or when the goy demands that the Jew be like him, in which case the issur is stricter.
According to the “nose-ring-is-assur point of view, a bigger problem is with human hair or synthetic sheitels that portend to imitate goyish hairstyles. After all, there can be no real Jewish hair styles for women, they cover their hair! Therefore, all sheitel styles are ipsi facto imitating the goyim and should be assur..
This way of thinking is of course nonsense, as no Jewish woman picks a wig style in order to “look like the goyim”. So too, I see nothing inherent in nose piercing that indicates that a woman who chooses to do so is doing so “to be like the goyim”. Same with almost everything we do, whether it be ear piercing, nose piercing, clothes, makeup, nailpolish or whatever else. It’s all permissible.