I don't think there's a tone you can take that will make non-technical users understand that it's "not your fault". To the non-technical user your site is just down and, potentially, a service they're paying for is unavailable.
I'd argue that such an outage _is_ your fault. If you're worried about Customer perceptions as a result of outages of third-party services your site relies on then, I'd argue, you need to have redundancy in your choice of third-party services. If it's important enough to worry about it should be important enough to spend some money on and do something about. If that drives up your costs then your product's cost, to your Customers, needs to reflect that.
I'd argue that such an outage _is_ your fault. If you're worried about Customer perceptions as a result of outages of third-party services your site relies on then, I'd argue, you need to have redundancy in your choice of third-party services. If it's important enough to worry about it should be important enough to spend some money on and do something about. If that drives up your costs then your product's cost, to your Customers, needs to reflect that.