Money is referenced fairly regularly in the original series. There was an episode when a supporting character (e.g., not Sulu, Scotty, etc.) had the helm and he remarked that he "bet Credits to navy beans" that an enemy was about to do something. The fellow who sold tribbles, in "The Trouble With Tribbles" had a considerable bar tab he'd run up. As mentioned, McCoy bribed someone credits to smuggle him to the Genesis Planet, in ST3. The list goes on and on.
Thats the thing. Its not money in the normal sense of someone minting the stuff and its in limited quantities. Its something you get for, well, doing stuff. Exactly what credits though is up for debate but they are not actual coin. At least not usually. TNG in particular suffered from writers changing stuff, because. So odds are somewhere in some episode someone IS paying with coin.
Outside the Federation all bets are off. Barter seems the main 'coin' of the realm. Find what someone wants in order to get something you want.
Money is a fixed unit of exchange to facilitate trading.
Well before coins we had money, sea-shells, cacao, feathers filled with gold dust just to name a few examples.
If we phased out all printed/minted money in favor of plastic debit cards ... Would that mean we no longer use money?
If we phased out all credit cards in favor of the chips implanted in our arms, would that mean we no longer have a credit system?
In México we are so smart we now use plastic bills, money made of plastic, harder to counterfeit, but it's neither coin nor paper, is it not money?
And all this is very relevant in RPGs, because I have seen time and again the missunderstanding of what constitutes both money and progres in futuristic settings:
Progress isn't going back to an inferior form of trading we call bartering. You want some eggs but need to sell a cow, I want some pork but only have eggs and the stuff only gets more complicated.
How many chickens to buy a cow? How many pigs? How many eggs to buy a quarter of milk? And if it's butter?
Money is easier to trade, to carry and thus much superior and evolved than bartering.