I can tell you for Arena Overlord, you just need to be #1 at some point. I just got Six Paths for the first time last month. And I was the first one on our server to get to six paths that month. But, at the end of the month I was not #1.

Oh, and here are some more for your list:

Hokage Encyclopedia - Acquire Full score on Ninja Test 10 times

King of Ramen - Owns 9999 bowls of Ramen

My idea on this is that it isn't just counting total wins, but rather the wins you get in row after the first. So, say you get 5 wins in a row, lose, then 3 wins in a row. Then it would count 4 from the 5 wins and 2 from the 3 wins, so you would get 6 added to your total. This is what makes the most sense to me for the wording. My level camping account has 163 total and I don't think I've ever gone through a sage battlefield without at least 1 loss. So, a loss doesn't reset the counter. The only question is if that first win counts to your total. I will try to check how the number changes the next time I do Sage.

Hmm, so none of this is true. I got 4 wins, one loss, 6 wins, one loss, 2 wins, 2 losses, end sage world in battle with no win streak, but max win streak of 6. I got no points toward consecutive win total. But, I've never finished without a loss, so my though now is maybe this number gets increased by the number of consecutive wins you have at the end of battle.

So, this time, I got highest streak of 8 wins, and ended on a streak of 3 wins. My number didn't change. So, it seems that the sage world victory streak counts between battles, but a loss at any time will reset your current counter back to zero. The number in reputation is your longest streak without a single loss across multiple sage battlefields. They have to have been counting for quite a while, because the only time I could have gotten 163 wins in a row is back when I was less than level 70.

well, I had many losses, but also won against streaks of 3, 3, 2, 6 (with losses in between). My points went up by 12. Which tells me that you get points for the length of the streak you end, but it has to be 3 or more and make that system message saying you broke players streak.

Yes, but their streak has to be a minimum of 3 to get points. breaking someone's 1 or 2 win streaks gets you nothing.