The term "bitt" appears in the Colonial Records of Pennsylvania in the 17th century, many decades before the introduction of arguably round pieces of 8 in 1732.
Prior to which, rough "cob" cons were minted. A "bit" or "bitt" is simply a small coin (similar to the British "threepenny bit"

, & was appleid then to the Spanish-American real since it was the smallest coin in current use. It has nothing to do with cutting of coins; this is a false etymology. Cut coins were strictly called "sharpshins.