Once you pay the fine uncontested, it's over. It's an admission of guilt. You are right the case is closed and unappealable.
If you wanted to contest it you have to either submit a TBD request or go to the arraignment (or possibly the court will take a not guilty plea by other means) and go to court.
"Calling the city" does nothing. Your case is heard in court for the appropriate county and even they won't intercede short of the trial (TBD or otherwise).
Looking at the intersection you have a no right turn on red sign. My guess is you didn't come to a complete stop before crossing the limit line (which in this case is the further east of the two yellow crosswalk lines) or the light turning green. As Mr. K alludes, the video will likely be telling.
For example in this google shot of that intersection:
https://www.google.com/maps/@34.1883...8i6656!6m1!1e1
The white car clearly appears to have run the red arrow. If the light hadn't been green, the black car also is in violation.

