I have a circa 1927 L.C. Smith 12 ga. Ideal Grade, that has Hunter One Trigger on trigger plate housing next to trigger, but has a Miller One Trigger. The safety is as you described.
Why the selector was changed to along side the trigger, like the Hunter One Trigger, was in my opinion foolish. At least the previous way you could select a trigger without fumbling around underneath.
In looking at the mechanics of a HOT, I can understand why that type is set up the way it is, but not for the Miller.

David