If you're able to do a successful rush pre-Feudalism you've already "won" and that's what I'm calling the situations where NTB makes "converting" the win easier than actually winning.
Again, as far as barbs go, I'm an extreme case, but I'm also thinking in terms of score and time, not just overall ease of victory. It may well be "easier" to win on raging barbs than no barbs now, but your score and time are always going to be better on no barbs.
The core of my argument, in a nutshell, is this: no barbs makes the "pre-winning" stage go
faster, while NTB makes the "post-winning" stage
easier. Since easier could perhaps mean a better score and/or time, I'd think it fair to add a modifier like the one that exists for no barbs.
I realize my way of thinking is a little bit different, but I "learned" civ 4 through the CS sling. Pulling it is effectively winning; understanding what you're doing to get there and then how to use the advantage generated is basically knowing how to set up a won position and then convert it. It extends to other parts of the game; the CS sling is/was just a blatant example of a won position. A more subtle won position is a successful conquest of a neighbor: once the budget is balanced, the player has double the production of the nearest AI and a recovered GNP so the process can be repeated - take over another neighbor, balance the budget - until it the game is done. The invasions happen in separate steps, but only the first one is when the player actually "wins"; it gives the player a decisive advantage (eventual double production), while the rest are just finishing the job. A substantial tech lead is a won position, too; converting it by, say, slinging Liberalism for Replaceable Parts is the first part of post-winning, researching Rifling gives you superior military technology and then conquering a world of medieval units with Riflemen is actually the very last part of post-winning.
As for advanced start; yeah, that's obvious. The AI is better, but its intelligence is still...artificial.

Sulla gives a perfect example of what I'm talking about with "pre-winning", a won position and "post-winning": pre-winning takes place during the analysis of the advanced start, the won position is turn one, and the rest of the game is now post-winning.
