Is the customer told automatically when I cancel their order?

No. Changing an order to Cancelled doesn't send the customer an email. They'll see the new status when they open My Orders on your store.

So let the customer know yourself, by email, phone or WhatsApp, especially if they paid online and need a refund through your payment provider.

Then record it on the order. Pick the cancellation reason (for example Items unavailable) before you click Update Status, and add a note such as "Customer informed by phone" with Add note, so your team can see it was handled.