There might be another angle (sorry for the double entendre!) here... which is - "what are you hoping for once the fish is hooked"? So the matchman, having hooked his fish (lets say, an 8lb carp) - he wants that fish in with the minimum fuss and struggle. So from what I've seen, that means playing the fish with the rod tip close to the water (horizontal as I think Mike puts it) where maximum side strain is brought to the fore and often the fish can actually be reeled in... and if it swims to the other side, then the rod is quickly brought over to exert pressure the other way, a bit like tacking on a sail boat. All of this is performed with the one clear objective of putting the fish in the net at the earliest opportunity with as little drama as possible.
However, for the pleasure angler - the fight once the fish has been hooked is the only reason he got out of bed at 6am this morning. Drama is everything, so likely he will hold the rod up higher (generally with carp I tend to find that makes them pull harder, I think it's because the fight becomes more three dimensional because he can swim left, right or down). If the fish wants a bit of line of the clutch, so be it. "look at him pull!" says the angler with glee.
I tend to use a bit of both to be honest. I do tend to find that getting the rod low and applying only sidestrain tends to put you in control of the fight quicker. But I have gone to have fun and I want the fight to have little drama.