Yup, I was right: the same stuff that counts for Challenges also counts for War. The key here is that it says "points", which the last 4 items in the War/Challenge report are not (they are... "counts", I guess?).
"The Tenth Wave" explains it better, but that's all that it is: an extra explanation (hence the parentheses). "Captains of the Black Flag" simply lacks the extra details, but it counts in exactly the same manner.
Before you say that it's inconsistent and/or confusing, I agree, but that's how it is and there isn't much that we can do about it.