Yes, you are correct. The word “negative” usually means the negative example (off-diagonal), and not the less than zero value. But there are inconsistencies when word “negative” is used and sometimes it is used to describe not just the negative example but also the less than zero value. Actually, now I think the quiz explanation should be corrected as you both noted to be more consistent with the course material.
Regarding you question about hard mining I would suggest to read my explanation to the same question:
If you still have questions, please feel free to ask. Cheers 