It sounds like you've never ridden a motorcycle? Most safety courses on motorcycles teach counter-steering, precisely because it is a thing and failure to understand it can compromise your ability to recover from a bad situation.
On a motorcycle going fast, you counter-steer continuously. If you make a slow right turn on a freeway, you can turn for 30 seconds by only pushing forward on the right handle.
I'm pretty sure you always counter-steer even on a bike, it's just very hard to notice, especially going slow. We don't think about it. It's not really a choice though, there's only one way to turn without leaning, you don't get to pick counter-steering or not.
You should definitely try it on a bicycle while going fast. Once you consciously turn your handlebars right and end up steering right no matter what you do, it becomes really clear.