Is "balanced field" defined with or without an engine failure?
It refers to the case with an engine failure.

The balanced field length is exactly equal to the accelerate-stop distance and the accelerate-go distance, if the V1 speed is chosen, where those two distances meet:


  
  
  2.3.1.3 Balanced Field Defined
  
  [...]
  
  The Continued Takeoff—After an engine
  failure during the takeoff roll, the airplane
  must continue to accelerate on the remaining
  engine(s), lift off and reach V2 speed at 35 feet.
  The later in the takeoff roll that the engine
  fails, the heavier the airplane can be and still
  gain enough speed to meet this requirement.
  For the engine failure occurring approximately
  one second prior to V1, the relationship of the
  allowable engine-out go takeoff weight to V1
  would be as shown by the “Continued Takeoff”
  line in Figure 10. The higher the V1, the heavier
  the takeoff weight allowed.
  
  The Rejected Takeoff— On the stop side of the
  equation, the V1/weight trade has the opposite
  trend. The lower the V1, or the earlier in the
  takeoff roll the stop is initiated, the heavier the
  airplane can be, as indicated by the “Rejected
  Takeoff” line in Figure 10.
  
  The point at which the “Continued and Rejected
  Takeoff” lines intersect is of special interest. It
  defines what is called a “Balanced Field Limit” takeoff.
  The name “Balanced Field” refers to
  the fact that the accelerate-go performance
  required is exactly equal to (or “balances”)
  the accelerate-stop performance required.


(FAA Pilot Guide to Takeoff Safety, emphasis mine)

The continue case (accelerate-go) assumes an engine failure one second prior to V1.