People can call themselves whatever they want, consider themselves whatever they want, live their lives however they want. Within reason.
How other people look at them based on facts, their own and generally accepted definition of what is, is another matter. The validity of something is not necessarily based on fact but on general acceptance. For example, the concept of the existence of ghosts. many people believe in them so much that is has become a part of human culture. But it does not necessarily mean they are actually real. Perhaps one day the term "flexitarian" will be popularly acceptable, perhaps not. But people who call themselves "flexitarian" have the right to do so whether you agree with it or not.
the term "vegetarian" itself was a relatively recent term. I think it was coined only in the late 1800s. I'm sure it met a lot of non-acceptance and even ridicule back then too.