I haven't played for a few years but back when I did (Urza block) you could respond to the damage going on the stack by paying 2 mana and then the 1 damage is considered the "next 1 damage", it's prevented.
I heard in later edition(s) the damage doesn't go on the stack or something so I guess that wouldn't work.
Wouldn't it make more sense to pay the 2 mana during your Untap phase and avoid the stack altogether?
At the beginning of your Upkeep the damage trigger goes on the stack. Then, both players have the opportunity to play spells and abilities. The Barbed Wire's controller can then activate the second ability and pay the 2 while the trigger is on the stack to prevent the damage when the trigger resolves.
No player gets priority during the untap step, so you couldn't pay for the ability then. The earliest time you'll get to play it is while the trigger's on the stack.
As far as damage on the stack is concerned, that change was for combat damage, which is resolved immediately at the beginning of the combat damage step
*Edited for nitpickyness