Honesty parking

In rural areas I've noticed a lot of small parking lots that appear to follow the "honesty" model. A small box is installed where you're expected to put in the required fee for your parking usage.

Here's an example:
https://goo.gl/maps/kcwEZqKopKcq2nq26

How does this work? Maybe I'm using the wrong keywords, but it's hard to find anything on it.

My main question would be: pay on arrival, or pay when leaving? Anything else I should know? (in general I mean, not specifically on the above linked one, that one is just an example).

I don't think there is a general rule that works for all similar lots around the country. More precisely, there are tons of places I can't figure out the system of, and this is not just about parking but about all kinds of services. So, I'd usually try to find a worker to ask.

But while I know that a lot of gated parking lots charge you upon leaving, I think that's because they charge you per time. If the fee is fixed like the one we see in your link, then I see no reason you shouldn't pay in advance unless you've run out of change. Or maybe if I ask, the shopkeeper might say something like, "If you buy something from us, we'll wave the parking fee", like how it works at malls in the city.

So, all in all, if I were you, I'd ask, and if there isn't anyone to ask, I'd try to put in the exact amount of coins upon arrival, but if I don't have small change, I'll let them wait until I'm done shopping and about to leave.

Like you said, these systems rely on honesty. I'd try to pay in advance, also because I feel that I could forget to pay on my way out. And I'd also try to communicate with the people to show my honesty.

I'm a local Japanese resident and traveler, if that helps.