Australia

For clarity: your idea of $10/year as a "nominal" amount is itself patently absurd. That's an enormous cost, many hundreds of times more than the amount that would be proportional to how much damage they do to the infrastructure, compared to other vehicles.

If you wanted a truly "fair" price, it would be measured in cents, if that. And at that point, the cost to administer the system would far outweigh the revenue brought in.

So tax cyclists a nominal amount. $10/year.

Absolutely fucking terrible idea.

Where would a cyclist’s compensation come from

From the driver. When bikes and cars collide, the driver is basically always responsible, so they're the ones who pay.

if they’re using the road, they should share the cost

Cyclists don't create a cost. More cyclists is literally a net positive economically, the exact opposite of car drivers. Cyclists cause negligible wear and tear on the roads, even once you account for the risk of crashes (which is actually a car's fault anyway), they're a lower burden on public healthcare, and they're more likely to be spending money at local businesses.

Literally everything about encouraging more people to ride rather than drive is positive. And by extension, putting up any barriers is a terrible idea. Even a "nominal amount" would deter huge numbers of people from cycling. And that's not the effect we want.