Let $C/\mathbb{Q}$ be a curve. I am trying to understand the difference between
(1) There is a model $\mathcal{C}/\mathbb{Z}$ of $C$ such that the fiber $\mathcal{C}_p$ is regular
and
(2) $C$ has good reduction at $p$.
According to Liu's definition ("Algebraic Geometry and Arithmetic Curves", Definition 10.1.19), (2) means that $C$ admits a smooth model $\mathcal{C}'$ over $\mathbb{Z}_{(p)}$. How should I understand this exactly, i.e., how is this related to a model $\mathcal{C}$ of $C$ over $\mathbb{Z}$? And does (1) or (2) imply the other? Any help is appreciated.
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