'At the wheel' means that you are driving. It's an adverbial phrase - almost a kind of idiom - meaning that you are in control of the vehicle. I have never heard anyone talking about being 'on the wheel'. Where did you hear this?

Your definition of "at the wheel" is correct. "On the wheel" is incorrect grammar, and I've never heard that expression in the U.S.

Thank you Matthew

In terms of driving - 'At' the wheel is used. 'On' the wheel would be like physically putting your hands or an object on the wheel of the car. For example: 'I'm at the wheel right now, so I can't answer the phone' 'He fell asleep at the wheel so he got into a car accident.' vs. 'It's important to keep your hands on the wheel at all times when you drive' 'I'm going to rest my head on the wheel because I'm tired' 'At the wheel' also has an idiomatic meaning as well as in 'being in control of the wheel' or 'being in control of the situation': 'Our group project did really well when Ali was at the wheel'