Protestant does mean something. It does have a meaning. It’s meaning comes from its origin, but it has also taken on the contemporary meaning of, simply, western non-Catholic Christian. Beyond these two things, not much of a meaning.
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As for wanting to be called, “the Church”, it is not only understandable, but accurate.* As for wanting to be called Catholic, not all feel that way, and wish to avoid the title, while for some, such as myself, it has always been and remains the correct title. Wanting it or not wanting it is irrelevant. Not all Catholics are in communion with the Bishop or Rome, though many are.
Jon