Douglas does not say anything about his definition referring to one trade, but rather to the intentionally vague "thing". His edge could refer to superior equipment (probability of my getting filled is higher than your getting filled) or to 5 trades of your trader (probability of winning after 5 trades is higher than losing after 5 trades).
If he was referring to the "thing" then I agree that it was exceptionally vague. ("It depends on how you define 'thing'." Hmm, reminiscent of a certain presidential testimony.) How useful. Perhaps everytick said it best and most succinctly with a legitimate degree of vagueness (and no more) in his post on the first page of this thread. Perhaps we should have just left it at that.
Actually, this is not the case. Douglas goes to great lengths explaining how to develop a high-probability/low-risk system. The "thing" is whatever is the outcome one is looking for as a result of his study, testing, trading.