Yeah, it's confusing. The Thunderer is the Uberti-made gun with the Thunderer-style grip (a lot of people call it a birdshead grip, but technically a birdshead grip is just rounded in back without the Thunderer "hump"). The Thunderball is the Pietta-made version. The grips on the Thunderball are just slightly thinner.
Then there is the Thunderstorm. That is the name used for any gun (single action grip or Thunderer grip) with the low wide hammer and an action job. I wish they hadn't decided to use the name "Thunderstorm" for those guns because it does add confusion. You can have a Model P Thunderstorm, or a Thunderer Thunderstorm, or a Thunderball Thunderstorm, etc. And yes, the Thunderstorm was originally for mounted shooting because the low wide hammer was not legal for SASS, but now it is.