As a Cornishman, this is basically what's taught in the schools, yes. You live in a godforsaken suburb miles out of London but you pretend that, unlike us, you somehow live in a city. ;-)

"isle of dogs. Or parts of peckham. Or West Ham."

All of these places are in London. They have London postcodes, are inside the M25, and are administered by the Mayor of London.

Pretty much every major city worldwide has large suburbs that have their postcodes and are administered by the same municipal government, but they're clearly not included in the "European city center" designation. They are part of the city, but not part of the city center; and for a multimillion city the difference (and distances!) are significant.

They are all "inner" London.