After all, "or" by itself is usually used to mean "exclusive or" in English.
No, it isn't. None of several common reference sources support the reading that "or" is usually an exclusive choice in written or spoken English. I'll quote the first:
OR —used as a function word to indicate an alternative (coffee or tea) (sink or swim), the equivalent or substitutive character of two words or phrases (lessen or abate), or approximation or uncertainty (in five or six days)
In most cases, the language requires you to infer the exclusiviity of the choice based on context.
Should you need to offer someone an exclusive choice, or wish to minimize the chance of confusion, the solution is to use more of our existing words rather than attempting to coin another.
I would you like either coffee or tea with your dinner?
You may pay your bill by either check, cash, credit card, or a combination of all three.