And how are those two suggestions mutually exclusive?
The 'Or' operator does not mean mutually exclusive. That would be the 'Xor' operator.
I guess you didn't make it down this far:
Following this kind of common-sense intuition about "or", it is sometimes argued that in many natural languages, English included, the word "or" has an "exclusive" sense. The exclusive disjunction of a pair of propositions, (p, q), is supposed to mean that p is true or q is true, but not both. For example, it might be argued that the normal intention of a statement like "You may have coffee, or you may have tea" is to stipulate that exactly one of the conditions can be true.
And they would be sometimes wrong. After all, why should you not be allowed to have coffee and tea?
So, we have established that in common English 'or' can sometimes mean 'XOR' and sometimes 'OR', thus rendering your initial objection irrelevant, thanks. I don't know how good your English is, but here's an example:
XOR: Would you like to go to the mall, or get groceries?
OR: Would you like to go to the mall or get groceries?
There's a matter of inflection apart from the comma which I can't convey, of course, but hopefully you get the idea.
Hopefully this addresses the OP's questions, thank you.