At a hanging belay, one typically does not remain motionless. People move their legs around, either just while fidgeting or balancing, or bracing their legs against the rock, or aiders.
I cannot speak to any differences between climbing harnesses and safety harnesses.
There are lots of relevant differences between climbing and safety gear and posture, but you're spot on about motionlessness as the key element.
Even for safety-gear falls, suspension trauma is somewhat overhyped for "slip and your harness catches you" accidents. The natural reactions - fidget, adjust for comfort, move your legs if they feel tingly - increase the safe time a great deal, so the main risk is falling into a hard-to-recover spot and getting tired.
That said, construction incidents are far more likely than climbing falls to be worst-case events: a substantial fall paired with injury or unconsciousness. When that happens, the impact pulls the harness tight and the victim doesn't adjust it or otherwise move around at all. That's where the stats about trauma starting in minutes enter the picture.
I cannot speak to any differences between climbing harnesses and safety harnesses.