Genuinely close, and the most interesting argument in the opening. Two respectable plays: split with 13/11 24/23, or slot the 5-point with 13/11 6/5.
13/11 24/23Rollouts are close
The position after 13/11 24/23. The 23-point and 11-point are highlighted.
Why this play
Slotting means dropping a lone checker on your 5-point and hoping to cover it next roll. If it survives, you own the best point on the board on move two. If it is hit, you have lost ground and a tempo.
The direct shot number decides it. A blot on your 5-point is 20 pips from your opponent's back checkers, which is out of direct range, but their builders can reach it. The exposure is real but modest.
Splitting is the quieter play: it moves a back checker to the 23-point, starts the process of escaping, and gives your opponent an awkward decision about whether to attack.
Modern rollouts put the split marginally ahead at money play, but the gap is small enough that match score changes the answer. When you need a gammon you slot; when you are protecting a lead you split.
The mistake to avoid
The actual mistake with 2-1 is playing 13/11 13/12, which does neither thing. It builds nothing and starts no escape.
How settled is this? Rollouts are close. Either main option is defensible, and the match score should push you one way or the other.
Knowing the play and making it under pressure are different skills
Play a rated game and the computer reviews every decision you made afterwards, including this one. The dice are sealed before you move and verifiable after, so what the review measures is you.
What is the best play for an opening 2-1 in backgammon?
13/11 24/23. Genuinely close, and the most interesting argument in the opening. Two respectable plays: split with 13/11 24/23, or slot the 5-point with 13/11 6/5.
Why not play something else with 2-1?
The actual mistake with 2-1 is playing 13/11 13/12, which does neither thing. It builds nothing and starts no escape.