Is it better to shut down or sleep my MacBook Air for optimal battery health?

Neither option affects battery health, but there is no compelling reason to routinely shut down a Mac.

Macs work best if you treat them with benign neglect. When you're not actively using a Mac, just ignore it like you would an iPhone: put it down, toss it aside, whatever. It doesn't matter. The battery and charging technology is practically identical.

Neither option will substantially affect battery health. I'd recommend just letting it sleep.

Regards.

Neither action affects the battery, but FWIW, I shut down my computer whenever I'll be away from it for more than an hour or so. No particular reason but I also shut down my TV, lamps, etc. whenever I am not using them.