Firstly, I must admit I am not an expert on the subject. Though, the model of movement for the leg-break and the leg-spin is the same, there are some differences.
The Leg Break is bowled with the help of both the movement of the wrist and the roll of the fingers using the anti-clockwise effect to induce extra spin whereas the Leg Cutter has the wrist firmly in position while still rolling the ball of the fingers to induce spin(not as much as the break) to cause movement off the pitch.
The Leg Cutter is generally used as a variation ball in a fast bowler's weaponry as a more effective version of the slower ball which results in the ball not only unexpectedly coming at a slower pace but also moving off the pitch, making it a very deceiving ball if bowled well. The Leg cutter can be bowled with only a slight loss in pace but for the Leg Break, due to the wrist movement and the increased roll of the fingers, a great amount of pace has to be sacrificed.
My take, on hearing about the great movement Barnes was able to get with the Barnes ball and the tales of wonder-struck batsmen is that Barnes bowled a genuine leg-break(a la Afridi, Warne) as his variation ball.
EDIT:- Sorry for the unintentional Hijack, Stephen.