Ah yeah there are multiple definitions of 'acceleration' here. Unruh radiation occurs when you're not 'in an inertial reference frame,' loosely meaning that you feel acceleration. So in a rocket in space or (presumably) standing on Earth's surface.

What you say makes intuitive sense, but it was actually the opposite logic that lead Einstein to his general theory of relativity. Here's a slightly dorky but very good Veritasium video that explains this issue and general relativity https://youtu.be/XRr1kaXKBsU?si=1iudoAx5kWgWHHt-