For your 'true horizon' to be 60 miles then you will need to be stood at 2400 feet above sea level (about 3/4 of the way up Snowdon

), looking across the sea.
The 'visible horizon' will be much less as it's used to describe the horizon when objects like trees, buildings, hills etc obscure to 'true horizon'.
The 'visibilty' may well have been 60 miles or more from your location, but trying to judge the distance based on the edge of a cloud layer that could be thousands of feet above sea level is going to be very difficult because you're looking at the edge of the cloud at an angle, not horizontally. So the edge of the cloud will be much further away from you than the visible or true horizon.
If that all makes sence?