ChristiaanJ, you have defined lift to be different from the rest of us. Here is what you said:
"Lift is the force acting on the aircraft at right angles to the air speed vector."
And since the airspeed vector is not level for a climb, your definition of lift has a backward cant that is equal to the climb angle. However, in fact, lift is typically defined as the vertical component of the force produced by the wing, ie in a direction straight up, regardless of the airspeed vector. The force produced by the wing will normally have a rearward component, which is equal to the induced drag, and which is included as part of the total Drag.
ft is correct, and Keith's case of taking the situation to it's extreme also provides a simple example that supports it.
Lift force is vertical, opposite to weight. And since we're talking about an equilibrium situation, and thrust has a vertical component, then lift must be less that weight by the vertical contribution of the thrust.