Destroying a permanent with ability "as long as this card is on the battlefield"
Provided that Player 2 sequences correctly, the artifact will cease being a creature before combat damage is dealt.

The situation that you're describing would play out like this:


Player 1 declares attackers, including the artifact creature.    
After attackers are declared, both players will receive priority. Player 2 can use their instant now to kill the Skilled Animator. At this point, the artifact is no longer a creature. This will cause it to be removed from combat, per rule 506.4 (emphasis mine):  



  A permanent is removed from combat if it leaves the battlefield, if its controller changes, if it phases out, if an effect specifically removes it from combat, if it’s a planeswalker that’s being attacked and stops being a planeswalker, or if it’s an attacking or blocking creature that regenerates (see rule 701.12) or stops being a creature. A creature that’s removed from combat stops being an attacking, blocking, blocked, and/or unblocked creature. A planeswalker that’s removed from combat stops being attacked.



The declare blockers step begins. After Player 2 declares their blockers (if any), both players will receive priority again. At this point, Player 2 can use their instant again, with the same effect.  
The combat damage step begins, and damage is dealt. After damage is dealt, both players will receive priority again. At this point, Player 2 can use their instant; however, at this point, it's too late to prevent the damage, as it has already been dealt.