The situation between .38 Special - .357 Magnum and .44 Special - .44 Magnum demonstrates the practice of 'developing' a 'new' cartridge from an 'old' cartridge.
The difference between .223 Remington and 5.56x45 NATO is closer to the confusion between .38 Automatic - Super .38 or 9mm Parabellum - 9mm Glisenti. Those two instances featured rounds with identical cases and bullets, loaded to significantly different pressure levels.
Red Dog, I'm sorry to say this, but explaining the difference in calibers is nothing but technicality. But good luck in your attempt.