Question

What is the definition of a "success" in a binomial setting?

Answer 1

In a Binomial setting, there are only two possible outcomes per try. Depending on what you want, you call one of the possibilities Fail and the other one Succes.

Example :
You may call rolling a 6 with a die Succes, and a non-6 a Fail. Depending on the conditions of the game, rolling a 6 may cost you money, and you may want to reverse the terms.

In short:
There are only two possible outcomes per try, and you may name them as you want: White-Black, Heads-Tails, whatever.
Usually the one you use as `P` in calculations is called (probability of) Succes.