Yes. Both buckets pull down on the hand with the same force, but the broad handle spreads that force over a larger area of the palm and fingers, so the pressure on the skin is smaller.
A narrow handle touches the hand along a thin line. The whole weight of the bucket is delivered through that thin strip, so the pressure there is very high and it bites into the skin. A broad handle rests across the whole width of the fingers. The weight is the same, the area is several times larger, and the pressure falls in the same proportion.
The round cloth pad that people place under a pot or a vegetable basket on the head (Fig. 6.3) works in exactly the same way. Without it, the rim of the pot presses on a small ring of the scalp; with it, the load is spread over a much wider patch. In both cases the aim is the same — reduce the pressure by increasing the area over which the weight acts.