I think it should be implicated instead. while Wikipedia and Dictionary.com both agree it can be broadly defined as a hairclip, Oxford specifies it further as a "typically bar-shaped clip or ornament for the hair." an Amazon search returns some very bar-shaped results, differing from the traditional thin, rectangular hairclip like post #4363816. if you scroll enough you'll also find some alternatively-shaped barrettes, which are more commonly seen. they’re very similar to the ones Higuchi Madoka uses, for example.
a cursory search throughout barrette and you'll find a bunch of images matching these descriptions, like post #1721293, post #4058386, post #4204720, post #4196878, post #4134900 and post #4058339.
I think the difference between barrette and the more general hairclip is great enough to warrant its own tag, but it’s still a subset, hence the implication.
Updated by John Fantasy XIV