The different names of Linux 'versions' are kernels, right?

Not really, the kernel is basically the only thing all distros share in common. The different Linux "distros" (as in distributions) are called that precisely because they have different ways of distributing the software they use to build the OS. They use different means to achieve the same goal.

Any recommendations on what 'version' to start with for a heavy gamer?

I am biased towards Fedora since I am a contributor of the project, but you can't go wrong with Ubuntu either. It's more popular so there's a lot more content about it online if you need help.

The only thing I recommend against is using small, niche distros or DIY distros (like Arch) as your first. Stick with big distros with big communities and don't be afraid to ask questions in the forums if there's any doubts or if you face any issues.

It sounds daunting in the beginning, but it's way easier than the internet make it seem.