When a Hero picks up gems or coins or other "collectibles" that add to his wealth in a game, what are those things (gems and coins etc) commonly called and how are they generally referred to in game design terminology?
You should distinguish from a heirarchical set of terms.
Loot is the overarching term from anything you would retrieve from a body or a reward for doing a exploration task (like the chest at the bottom of a dungeon). Sometimes quest rewards fall into this definition but are usually different.
A sub-category of loot is Currency. Currency is anything that you can spend to acquire something else, but are generally not useful on their own. One classic currency are Justice Points in WoW, which evolved out of Badge of Justice, looted from dead bosses in heroic dungeons. Some currencies are tradable, others are not.
A sub-category of currency is Money. Money is, specifically, something you can use to trade with other players or the game to acquire goods. Not all currencies are money, as illustrated above. If it isn't tradable to other players, it probably isn't money. Inherently, money exists as a medium of exchange in a market, allowing market prices to dominate. Thus, if the prices are set, even if it's called gold, it's more likely to be considered generic "currency" than it is money.
I would typically refer to them, as you said, as collectibles. You could also refer to them as tokens, but that may be confused with additional tries.
Collectibles might also refer to other things as well, such as items in an rpg or even overarching level goals. To refer specifically to the type of collectible that's picked up in a level and usually gives something special at 100, you could try referring to them as coins, typically from Mario, but that could get people confused if they're not actually coins. Another term is goodies, but that could be considered too vague, as well.
I would refer to them as currency. You've said yourself it represents wealth. If these are not actual "collectibles" in the sense of a trophy/etc unique item, and especially if they can be bartered for items and the like within the game, currency makes the most sense to me. It works for gems, it works for coins, it works for seashells, it works for bottlecaps... all of the above can be used as currency in the appropriate games :)
Game design is not a rigorous field when it comes to terminology. What we use in the field is more of an ad hoc vernacular than anything else. Often existing English terms capture our use cases just fine, so that's all we use.
This is one of those cases. You'll find as many ways to describe these collectible items as you could probably think of. "Collectibles," "pick-ups," "rewards," "chits," "tokens," "currency," et cetera. While the terms may not vary much within one game, they will across many. The initial term will likely be coined based on the specifics of whatever the designer or designers want to convey about the purpose of the item in the game.
It depends on what you plan on doing with them.
- If you simply collect them and store them as an 'award' or, in your situation, you stack them and it represents your wealth, you can use the term collectible.
- If the player will consume them to buy other stuff, i.e. if it used as currency, you should probably refer to it as, well, currency.
However I don't think there is a common way to name them across all game companies. You should use common sense and what works best for you and your team internally.
What's important though is that what the player sees is consistent across the board: don't use 'collectible' on a specific pane, and 'currency' on another, this would create confusion (you'll probably even end up giving it a specific name like 'Coin' and use only that term everywhere).