Does Turkish have two separate words for OR? 

  In English, OR can be either:

    1) One of either A or B but not both (exclusive OR)

    2) Either A or B, or both (inclusive OR)

Because of the ambiguity, the dumb answer is sometimes given:

    "Do you want to eat or not?"  ->  "Yes."

     Which of course is not an answer and the question has to be restated.

Turkish has two main words used for or

veya, ya da (I believe both can be used, with no difference in meaning)

yoksa (or, otherwise, if not) can also be used in questions to mean ´or´

click here for a small discussion about ´or´

For several choices with ´or´, you can use

ya X ya Y ya da Z (either X or Y or Z)

I have also seen ve/veya used to mean and/or for your example of the unusual inclusive meaning of ´or´