Let's say you pay x in the first scenario and y in the second, and normalise the utilities so that U(death) = -1 and U(alive with your current wealth) = U(0) = 0. Then case 1 yields expected utility either -2/3 or -1/2 + 1/2 U(-x) and case 2 has either -1/3 or U(-y). On assumption that utilities are linear in money, U(z) = qz, a rational agent will pay the solutions to the linear equations

1. -2/3 = -1/2 - qx/2 , x = 1/(3q)
2. -1/3 = - qy, y = 1/(3q)

Thus, the standard utilitarian reasoning gives that you should pay the same and the linked argument is correct.

The incorrect argument relies on a bit different calculation: it includes the negative utility form lost money even in the case you die. In such a case (assuming the disutility of lost money simply adds to the disutility of death), we get

1. -2/3 = -1/2 - qx , x = 1/(6q)
2. -1/3 = - qy, y = 1/(3q)

paying twice as much in case 2. The problem is that counting monetary loss after death as additional disutility contradicts the explicit assumptions (no heirs and all money disappearing after your death) and a reasonable implicit assumption (you have no chance to spend money after you realise your participation in the roulette and before the outcome is known).

Edit: the assumption of linear utilities is not necessary for the conclusion that you should pay the same.