@ProfDrHack : I solved a few hundred puzzles on Lichess and I never had one with the scenario which you describe here.
On
chesstempo.com I had several of these positions where alternative solutions existed. If you try one of these alternative solutions, the computer simply tells you that there is a better solution and you can then try to find it. You didn't fail.
Every line which ends with an advantage of more than +2 is considered an alternative line.
I had some scenarios on Lichess where you have 2 good candidate moves which seem to be winning. Sometimes it is hard to decide which one is correct. In all of these examples I later found out that the alternative line gave you only an advantage of +1.5, while the best line was +3 or better. These puzzles are the hardest for me to solve. But you learn a lot from those examples because you can then try to find out why the computer comes to this evaluation.
@jmss0906: You write that there are puzzles with an alternative second mate and that you fail if you try this line. To be honest, I don't believe that you can come up with examples for this. I also had puzzles on Lichess where I thought that I found a mistake and the puzzle must be wrong. Later I always found out that I was wrong. I simply had missed something very important in the position. For example, a few days ago I thought that I had found a puzzle which must end in a draw. Even after 30 minutes, I was not able to spot the correct answer. Then I realized that I can simple capture a pawn en passant. I missed this because I didn't check the move list.