Why should small o notation has to satisfy the equation for all values of the constant
Consider this also (more?) common definition:

$\qquad\displaystyle f \in o(g) \quad\mathbin{:\!\!\iff}\quad \lim_{n \to \infty} \frac{f(n)}{g(n)} = 0$.

Do you see now why we need to get below all factors $c$ in your definition? If not, we would have $f \in \Omega(g)$.