"Edge", whether it's House Edge or player Edge refers to the amount of expected gain or loss per unit wagered, or per amount wagered.
In dulence's example, he says you can expect to win one double up and lose one double up. This means that you will wager £200 on the double up you lose, and you will wager £4,000 on the double up you win (and subsequently grind out) = £4,200. On the double up you win, you will start with a balance of £400. Since you have already wagered £200 on the double up, you only have £3,800 WR left to do, and at a HE of 0.5% you will expect to lose £19 during the course of the wagering.
Thus your expected gain from the double up you win is £281. Since you lost your (-£100) on the first double up, your profit over the two offers = £181. To acheive that you have wagered £4,100.
Had you have grinded out the wagering in both cases, your EV would be £160, in £8,000 worth of wagering.