Let $\Omega$ be a bounded Lipschitz domain with boundary $\partial\Omega$.
There are two ways to define a space $H^1(\partial\Omega)$:
By using charts, we can define $H^1(\partial\Omega)$ to contain functions $u\colon \partial\Omega \to \mathbb{R}$ such that $u\circ g_i \in H^1(D_i)$ for all $i$ where $g_i\colon D_i \subset \mathbb{R}^{n-1} \to \mathbb{R}$ is a chart map. The norm is the obvious norm.
We can define a tangential gradient on $\partial\Omega$ as: $$\nabla_S \varphi = \nabla \varphi - (\nabla \varphi \cdot \nu)\nu$$ where $\nu$ is the unit normal vector on $\partial\Omega$ and $\nabla$ is the usual gradient. Here $\varphi$ is smooth. We can get a weak version of the tangential gradient by using the integration by parts formula on surface, let's call the weak tangential gradient $\nabla_T.$ Then we can define $H^1(\partial\Omega)$ as functions $u:\partial\Omega \to \mathbb{R}$ such that $u \in L^2(\partial\Omega)$ and $\nabla_T u \in L^2(\partial\Omega)$, and give it the obvious norm.
My question: are these definitions equivalent in some way? Do we have equivalence of norms? The second definition is not very common or popular, why is that?