This answer cites the Ralbag as interpreting a (critical-to-the-question) vav prefix on a verb as "or". I've heard before that a vav prefix is not always "and" and can be "or" or even "but". Is there any grammatical rule by which we can tell when to read it which way, or do all the non-"and" readings always arise from judgments about what would make most sense to the interpreter? Do Chazal generally agree on how any given vav is translated, making it received tradition and there are no explicit rules?
Vav does not necessarily mean and or or. It is a conjunction and translated according to context. I translate most vav's as a "soft then"- a subsequent event where formal written English would not put a then, but in a conversation, one might say "so then I..."
Other possibilities: but (although, however), while, since, therefore.