Is $0! = 1$ because there is only one way to do nothing? The proof for $0!=1$ was already asked at here. My question, yet, is a bit apart from the original question. I'm asking whether actually $0!=1$ is true because there is only one way to do nothing or just because of the way it's defined.

Yes, precisely there is a unique function $\emptyset \to \emptyset$ (with empty graph), which happens to be a bijection ($\operatorname{id}_\emptyset$).  Note, that $n!$ is the number of bijections $\{1,\dots, n\}\to \{1,\dots,n\}$.
