Use of "Or", inclusive or exclusive?

In English, or is always disjunctive. There is no ambiguity.

It is either one or else it is the other. It is not both. That is why when it could be both, one always says “one or the other or both”. The or does not admit the possibility that both may apply. So you have to add or both if you want both to be a possibility.

In other words, it corresponds to the computer’s xor operation, not to the computer’s or operation. The English or is really an xor.

As Bill notes in a comment, it is always grammatically disjunctive. So “Either Peter or Paul is coming to dinner with Mary” demands a singular verb, and you should set two places at the table, not three.

If you said that in chess, the queen can move any number of unblocked spaces either vertically or horizontally, you know that she cannot do both at the same time. However, with your own game, I cannot say, because I do not know its rules. The question comes down to whether you have to move in a straight line in the game you are playing, and I do not know your game.

It seems to me that how a person uses 'or' is very much defined by their perception of what they mean, rather than by any formal rules of the English language. And this can usually be interpreted from the context, particularly in speech, by a person with English as a first language.

And this gives rise to another problem - the difference between written and spoken language, and the problem of writing language as it is spoken, rather than having an awareness of the difference between the two (with spoken language usually having much more nuance, through changes of vocalisation, tone, etc., than written language, and a hearer usually more able to quickly discern a meaning).

Be that as it may, English is my only language and I have always regarded 'or' as exclusive, although I know that others do not do so. So when I mean one or the other or both I usually insert an 'and/or', particularly in written text.

Also, someone mentioned the use of 'either... or' somewhere, on this or another page. It seems to me that as soon as we use 'either' we are automatically exclusivising the 'or'. However, I don't think it's really the 'or' that's exclusive in this case, but rather the combination of the two words that gives rise to the exclusivity.

It is untrue that the natural English or is clearly and uniformly disjunctive. If it were, it would be impossible to add "or both" or "or whatever" to an or clause, or to make a list of more than two items, as in StoneyB's examples:

“Help yourself to chicken or ribs or chili or whatever strikes your fancy” and “I like Dickens better than Trollope or Scott or Thackeray”

RJB is correct that

how a person uses 'or' is very much defined by their perception of what they mean, rather than by any formal rules of the English language

Often we use or to convey a feeling of subjective doubt:

He speaks French, Spanish, Portuguese and German or Finnish, I forget...

The point is not "either G/or F"; the point is "maybe G, maybe F...". And while we tend to use "either.. or..." to strengthen the sense of "not and", this is a matter of degree not kind. There's a nice page on contract drafting here and StoneyB's superb coffee & cake example makes it clear that context is key, (or) perhaps even king. Note also that English uses or in negative sentences such as the following:

Bob's not in his room or his study.

Clearly in logical terms what is meant here is "Bob is not in his room and Bob is not in his study". Natural English does not use and because the conjunction falls within the scope of the negation (and it's obvious that Bob can't be in both places at once, so why say it?). In more formal English, we would say he is neither in his room nor in his study. Some languages, such as Chinese, use a positive conjunction and two negative particles to convey this idea (他不在房間也不在書房).

But I'd agree that, apart from contextual considerations, the default meaning of or does tend to be disjunctive. Frequently when translating the Chinese or, I have to use and in English because the or is insufficiently inclusive.