In fact, for unital $C^*$-algebras non-degeneracy just means $\pi(1) = 1$. In the non-unital case there is even a sharper statement than you tem (2): One can find for every $\phi$ and every $\epsilon > 0$ another vector $\psi$ and a positive positive algebra element $a \in \mathcal{A}^+$ with \begin{equation} \phi = \pi(a)\psi \quad \textrm{and} \quad \|\phi - \psi\| < \epsilon. \end{equation} This is nice as is shows that we do not just get a dense subspace and we get in some sense as close as possible to $\pi(1) = 1$. I found this in Blackadars encyclopedia book in Theorem II.5.3.7 and in II.6.1.5. Might be worth a look :)

In fact, for unital $C^*$-algebras non-degeneracy just means $\pi(1) = 1$. In the non-unital case there is even a sharper statement than your item (2): One can find for every $\phi$ and every $\epsilon > 0$ another vector $\psi$ and a positive algebra element $a \in \mathcal{A}^+$ with \begin{equation} \phi = \pi(a)\psi \quad \textrm{and} \quad \|\phi - \psi\| < \epsilon. \end{equation} This is nice as it shows that we do not just get a dense subspace and we get in some sense as close as possible to $\pi(1) = 1$. I found this in Blackadars encyclopedia book in Theorem II.5.3.7 and in II.6.1.5. Might be worth a look :)