If that's the intent, it's certainly poorly worded. If the intention is what you say it is, I would expect the rule to read: "A long rest is a period of extended downtime, at least 8 hours long, during which a character sleeps or performs light activity for no more than 2 hours" (so that "for no more than 2 hours" modifies "performs light activity"). By having "no more than 2 hours" in the sentence where it is, it's ambiguous about whether it modifies the nearest verb phrase ("standing watch") or the whole clause, but the bias for most readers of English would be to have it modify the nearest verb phrase rather than the whole group (though the ambiguity is certainly there).
Having said that - if there was an expectation that you needed to sleep for at least 6 hours during a long rest to benefit from it I'd expect to see that rule be more explicitly laid out somewhere rather than leaving it up to a DM to infer it from those sentences. If that's the expectation then it's another example of how that rule is really poorly worded.