To the OP...
I doubt in fact that any commercial ammunition for stalking or use for hunting (therefore with an expanding bullet of whatever style) will be in anything BUT boxes marked .223 Remington or casings headstamped .223" or .223" Rem or similar. Now the connected question is would an RFD sell you ammunition headstamped 5.56mm where your FAC says .223" Remington?
The same arises (although less so now as it isn't really much used outside of users of old military rifles) if an RFD would sell you 7.92 Mauser ammunition on an FAC for an 8x57 or 8mm Mauser. Back in the day it was never an issue. Nowadays of course most Nazi 7.92 Mauser ammuntion is s5 so it is a moot point. But yes before the Tory "armour piercing ammunition" ban I bought such 7.92 Mauser and also, indeed, 7.92mm BESA ammunition for use in an 8x57 stalking rifle I once owned in the early 1980s.
So yes a commonsense RFD should sell you ammunition for target use. But as you say do be aware that pressures are different!
The shooting catalogues are not exactly full of rifle cartridges with two names or alternative names but yes, there's .244 Remington and 6mm Remington, there's .303 British and (I'm looking at some fired cases) 7.7mm but as things have become less confused the 7.7mm designation by some nations of the .303 British has gone into history. True if anyone wants one I'll be listing them on SD Classifieds later this week and of course the classic USA vs Europe .25ACP aka 6.35mm, .32 ACP aka 7.65mm and .380 ACP aka 9mm Kurz.
Hope it helps! And whilst I'm here as a kid, as a teenager as someone on my age twenties, thirties and even forties it was always just ".303" never this American imported naming of ".303 British"!