K.S.,
You're right; what most people mean when they say "successfully continue the takeoff" includes a bunch of things, including achieving the required altitude by the departure end, required climb gradients, etc. Depending on the airplane, there can be lots of assumed things... "#1 windmilling on NTS, other 3 engines at takeoff power, gear down, flaps in takeoff position, 5 degrees of bank away from the dead engine, gear retraction initiated within 3 seconds, feathering the dead engine initiated within 6 seconds, 35' by departure end of runway, etc etc etc ad infinitum." A takeoff in a flat area might be considered "successful" even if you didn't quite make 35' above the departure end, but the engineering data uses that as the starting point, since departure procedures use that as the basis for obstacle clearance.
There is one instance which can occur where "balanced field" actually is a player: "how much weight can you take out of here?" Under certain circumstances, the answer will put you at exactly balanced field -- you can stop the airplane no faster than your V1, and you can successfully fly the airplane (minus an engine, with all the "assumed" things above) no sooner than V1. You have exactly balanced field, and they load up the airplane to exactly that weight. Any more weight & you'd have the dead zone of no option if you lose an engine between X and Y airspeed.
There are plenty of other times, though, when you aren't runway limited. Perhaps you are climb limited -- you have plenty of runway to stop or to go, but above some particular weight you can't clear the obstacles minus an engine. In that case, balanced field won't be much of a player.
Or perhaps the conditions at your landing field limit your max takeoff weight to something well below what would give you a balanced field, and so you don't have to perfectly reproduce what the test pilots did in order to survive an engine failure at V1. That's frequently the case if your max landing weight (for normal ops) is significantly below your max T/O weight.
Cheers,
Snoopy