My take on this is that there is just one possible interpretation of the sentence's meaning: You are not running, and the reason you're not running is because you don't like exercise. The other meaning that you gave - you are running but it is not because you hate exercise - although grammatical, is logically awkward: you are running and the reason you are running is because you do not dislike exercise. This is kind of hitting the listener over the head logically, like saying, you are running and the reason you're running is because you are not missing your legs.
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