FTA: “As a motivating example: player 1 (hereafter dubbed "Red") can win by playing in the center column on the first move and then following the weak solution's suggestions, but would not be guaranteed to win if the first disc is played elsewhere. The weak solution contains no information about what would happen in the other columns- As far as Red cares, it would be redundant to learn those branches, since they are not good.”

I don’t think that “since they are not good” is necessary for a weak solution. Even if every first move were winning, it still would be redundant to learn how to win for every possible opening move.

A weak solution gives you a guaranteed way to move from START to a win, whatever counterplay, not all ways to go from START to a win, whatever counterplay.