"Or" is ambiguous in daily life
So says @Arthur (and so say many others). But interestingly, this isn't the majority view among linguists, as I understand it. And (even if it is a bit tangential to the OP's actual question), maybe it is worth saying a bit about why.
You need a bit of key background. A standard approach to explicating how we manage to interpret what we read or hear is that the overall message conveyed is a result of the interaction of two things, first the semantic content (the "literal meaning") of the sentence used, and second contextual and pragmatic clues.
Now it is agreed on all sides, as others have been emphasizing, that when a speaker asserts something of the form A or B, sometimes the intended message is A or B though not both and sometimes the intended message is A or B or both. But -- given the standard approach -- it just doesn't follow from this that "or" in English is semantically ambiguous. For it could be that the literal meaning of "or" is inclusive disjunction, and when the overall message is understood (and/or is intended to be understood) as being exclusive, that is relying on additional pragmatic or contextual features of the occasion of utterance.
Note for example that in the sort of cases typically invoked to supposedly illustrate the uses of exclusive "or", it would -- on the semantic story -- be contradicting yourself then allow for “both". But this is not usually the case.
I say, for example, "Either there will be a peace treaty or the war will drag on another year." It is common background knowledge that peace treaties and continued war don't usually go together, so you rather naturally hear my claim as giving you exclusive alternatives. But of course it would be entirely coherent, I wouldn't be contradicting myself, if I continued "and such is the fragility of the situation, maybe both." It doesn't seem that I'm changing the literal meaning of my original claim, but just cancelling a pragmatic implication.
Note again that 'either ... or' in English seems to have a uniform semantic negation, 'neither ... nor ...' (which couldn't be the negation if 'or' is exclusive). And so it goes.
Now I'm not saying that arguments such as these are decisive (we've only scratched the surface). But I am saying it needs theoretical argument to determine which of the following views is correct: (1) "or" in English has two different literal meanings, vs (2) the literal meaning of "or" is unambiguously inclusive disjunction, and when there are intimations of exclusive alternatives these are due to pragmatic/contextual features. It is not obvious just from our linguistic behaviour which is the correct theoretical account of what is going on.