If it were me, I'd take them to the tribunal, because on your evidence you did not pass the sign on the left, but turned round via the triangular island. For me, this means the contravention did not occcur, but it's your money so if you decide to pay the discount, I understand.
?? The picture shows the OP leaving the zone having presumably passed the entrance signs.
No it doesn't. Look at the Google Street View of the location. OP says he turned into the street, saw the sign, and then turned round using the triangular traffic island, so at no time did he pass the sign on the left. The sign on the right cannot be seen by drivers turning left into the street.
That's not true. Look at the council picture provided, the OP's car is adjacent to the pedestrian dropped kerb, and, from Google Street View it shows that the sign is before the crossing when entering the road, so therefore they must have passed the sign. That's not to say they may have a de minimus case though, but, as Birmingham don't reoffer the discount, it's a gamble.
OP, don't expect a quick reply to your appeal. In my experience it takes at least a month to receive a response.