Sorry, I used the example of science because that is the flavor that OP used for their particular brand of Atheism, but you’re correct - technically you can believe there’s no god and science is wrong, though most people I’ve met who don’t believe in god do so because of their trust in science and their belief that it leaves no room for religion.

Again, though, a system of beliefs is not a system of behavior. You do not need a book or anything else telling you what to do for it to be a system of beliefs. Technically you could form a system of beliefs around the existence of frogs if you thought about them enough. You wouldn’t even need to change your understanding of frogs at all - just the act of thinking hard enough about them to fully understand and articulate your thoughts forms the system itself.

If an atheist burns a tree down because they’re an atheist, then obviously it does have to do with them being an atheist. Their brain made a connection between being an atheist and burning the tree down, and that’s the connection - it doesn’t matter if there’s an established set of rules they’re meant to follow or not. Just like how Christians can be expressly told to love their neighbor, and still use Christianity to defend their act of hating their neighbor - how people’s brains justify their reactions to their beliefs is their own system.