No, it means "Zero score at the time the job runs that awards the badges".
I guess this is how it works, sometimes timing is important:
Suppose you've got 30 Answers, 5 with 0 votes and 25 with upvotes. You don't get a Tenacious badge due to the "more than 5" rule.
Scenario 1:
- One of the 25 upvoted answers gets downvoted to zero score (at this point you qualify)
- The job that awards the badge runs: Badge awarded
- Your answer is upvoted again (or downvoted further), now you don't qualify anymore, but you keep the badge.
Scenario 2:
- One of the 25 upvoted answers gets downvoted to zero score (at this point you qualify)
- Your answer is upvoted again (or downvoted further), so you don't qualify anymore.
- The job that awards the badge runs. You don't get the badge awarded, even though you were elegible to get it somewhere inbetween.
Of course, the usual scenario would be to gain the badge for a newly accepted zero score answer instead of a downvoted one, but this kept the scenario of getting/not getting simpler I thought. The timing issue may or may not arise, this depends on how often the badge awarding jobs run. Since I've had badges that took a day to appear, this may come into play in some cases.