It's pretty clear that Harry knows what the palace means by the phrase private citizen: someone who has no official role serving society. MPs are public figures and I would argue, thinking as I type, that judges are public figures as well as Members of the working Royal family. Public figures serve the public. Harry serves himself.
Celebrities in contrast have a public profile, in the press or on social media. Celebrities come in all shapes and sizes, some through talent, like actors, some through social media, like influencers. You don't need a specific role or talent to be a celebrity, just a public profile.
Harry doesn't have to like this but if he is to have any peace he needs to accept that he is no longer a public figure but a private citizen with a celebrity profile. Being 5th in line to the throne is as irrelevant as being 35th and being the son of the monarch is not a public role. He stepped down from that section of the Mountbatten-Windsor family who are public figures.
Incidentally I agree with @My2cents1975 that he needs to be removed from the Royal website. I don't actually think it's indicative of anything more than the fact that the palace hasn't yet grasped the importance of a good website. The whole thing is a mess and not fit for purpose in the 21st century.