on-premises, not on-premise. but if these are hosted in AWS and aren't actually on-premises, maybe use of the term is not correct at all?

Yes, the premise with cloud is that the hardware is not located at your premises.

Tripped me up immediately. On-prem means I should be able to walk up to and point at the box. Worst case it’s in some data centre but it’s sure as heck not AWS

yeah that's not on-prem

On-premise, in the cloud, on someone else's premise. The future is now!

The software is on-premise, as in it runs on infrastructure you control, as opposed to a SaaS. Self-hosted might be the better term, but in the context of self-hosted GitHub runners, it’s a bit of a mouthful to say self-hosted solution for self-hosted runners.

It runs on infra you control, but it is objectively not infra that is on-premise. On-premise means "On my land, in my building, on the infrastructure I own." Self hosted is the correct term

If we already have some EKS clusters setup, is there a benefit to this over Actions Runner Controller (ARC)?

https://github.com/actions/actions-runner-controller

It’s going to be easier to maintain and cheaper. Plus you get compatible images included and more flexible runner assignment. But if you’re already happy with ARC then there is no real need to switch. RunsOn is really for people who want more of a turnkey solution.