I need clarification on the use of the word "either"

In two different text the first one I used to "combine" the elements in the "or" sentence in the other I only have 2 choices.

Both sentences are from boardgames: "Play if at least 3 Imperial systems contain EITHER a sabotage marker or a Rebel unit"

In my understanding this can be used (play) when I have 2 sabotage marker and 2 rebel units on Imperial systems.

The second one: "You may EITHER exchange your cards with another player or discard any number of cards and draw en equal number of cards"

In this case the rulebook clarify the ruling saying you "CHOOSE" one action.

I think the key is in the "or" placement in the second one you select actions in the first one items that can be combined.

If I'm wrong in my interpretation (I'm not English native) excuse me and thanks for any help in this matter if I'm correct: Is the wording in the first one ambiguous or bad wrote?

Thanks again

(In forums about rules of the first game the community say this is a correct use combining BUT my friend don't like it because the "either" word about choices I tell him is about "grammar" or phrase construction not about only that word, even some "people that knows English" say he is right but, as we, not English natives)

In Latin, "vel" is a non-exclusive "or" whereas "aut ... que" is an exclusive "or." English is not so clear-cut. Many people view "A or B' as ambiguous between exclusive and non-exclusive meanings. If you want to avoid risk of ambiguous interpretation, I suggest writing or saying:

(Non-exclusive "or") "A or B or both;"

(Exclusive "or") "Either A or else B."

I have known, however, careful speakers who intend an exclusive "or" when saying "either A or B," and who intend a non-exclusive "or" when saying "A or B." They may be strictly correct, but I prefer to avoid the possibility of misinterpretation.

As Jeff Morrow's answer says English does not distinguish between the "inclusive or" (both conditions can be true) and the "exclusive or" (only one of the conditions can be true), you have to work out which is meant from the context.

A good example of this is the question "Do you have a brother or sister?" The questioner could be asking whether the other person had a sibling at all in which case the answer could be "No", "Yes", "Yes, a brother (or sister)" or "Yes, lots of them"; this is the inclusive or.

However, if the questioner already knew that the person had one sibling, they would actually be asking "Is your sibling male or female?" This is an exclusive or (unless the person has an intersex or transgender sibling) and the answer needs to be either "a brother" or "a sister". "Yes" or "No" would usually be meaningless.

In your question and in the case of the second board game the "or" is definitely exclusive, if both actions were permitted during one turn the instructions would not say "or" they would say "and" or "then" to indicate that both actions are permitted during one turn.

In the first case there is a slight possibility of confusion if it is possible for a system to contain both a sabotage marker and a rebel unit.

If that is not possible then the nature of the "or" makes no difference. But if it is possible then an exclusive or would mean that a system with both a marker and a unit would not count towards the threshold. This is so unlikely that most native speakers would regard it as inclusive and say that one sytem with a marker, one with a unit and one would both would mean that you could play. However one could criticise the rules for leaving this possibility of confusion open.

It is important to recognise another difference between the two sets of rules. In the second game the "or" specifies an exclusive choice between two courses of action whereas in the first the "or" forms part of the conditions under which you can play.

Unfortunately there are no rules in English about when an "or" is inclusive or exclusive. You always have to look at the context and, if in doubt, ask for clarification.