Confusion about domains in PDE and "every $C^k$ manifold can be thought of as a smooth manifold"

Question

I have read that every $C^k$ domain can be described instead by $C^\infty$ chart maps. So in this sense every $C^k$ domain is a $C^\infty$ domain. But consider standard PDE where people say thing like "Let $\Omega$ be a bounded $C^1$ domain and consider on it the heat equation..."

In this case, $\partial\Omega$ is a $C^1$ hypersurface. Why not people just say "let $\Omega$ be a $C^\infty$ domain", if the chart maps can be smooth for every $C^1$ domain?