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Foreign coins being sanctioned currency for some time-- the U.S. did not unify its currency until the 1840s, and not till well after in the Wild West-- frontier Americans came to use "two bits" as slang for "one fourth," and so two bits is 25 cents, four bits is 50 cents, etc.
It didn't matter that 1 bit was 12.5 cents - sometimes chopping up a coin was the only choice. So they didn't think about the fine details - just what was possible with the technology (an axe or hatchet) and the coins at hand.
But, taking your idea into account, you won't hear John Wayne mention "one bit" in a monetary context. One bit applied to what he put in his horse's mouth, or how much he didn't care for something. :)
The tradition of cutting coins dates back almost as far as that of coins themselves - The non-circular shape of some coins and the serrations on the edges of others is intended to stop people from slicing around the rim to remove a small quantity of silver or gold, thus devaluing the coin.