Alexander Campbell's guess is correct.

Here is a reference. Lemma 10.2 of this paper https://arxiv.org/abs/1112.0040 shows that $Gaunt_n \simeq \tau_{\leq 0} Cat_{(\infty,n)}$. That is to say they are precisely the $(\infty,n)$-categories $G$ with the property that the space $Map(C,G)$ is homotopically discrete for all $C$.

They can also be described as the localization of $Cat_{(\infty,n)}$ at the single morphism $S^1 \times C_n \to C_n$, where $C_n$ is the free-walking $n$-cell, $S^1$ is the circle, and the map is projection. This description is model independent (for example the $n$-cell can be characterized model independently as in the proof of lemma 4.8 in the same paper above), however it is easiest to check that this description is correct in a particular model such as Rezk's $\Theta_n$-spaces.

Alexander Campbell's guess is correct.

Here is a reference. Lemma 10.2 of this paper

Clark Barwick, Christopher Schommer-Pries, On the Unicity of the Homotopy Theory of Higher Categories, arXiv:1112.0040

shows that $Gaunt_n \simeq \tau_{\leq 0} Cat_{(\infty,n)}$. That is to say they are precisely the $(\infty,n)$-categories $G$ with the property that the space $Map(C,G)$ is homotopically discrete for all $C$.

They can also be described as the localization of $Cat_{(\infty,n)}$ at the single morphism $S^1 \times C_n \to C_n$, where $C_n$ is the free-walking $n$-cell, $S^1$ is the circle, and the map is projection. This description is model independent (for example the $n$-cell can be characterized model independently as in the proof of Lemma 4.8 in the same paper above), however it is easiest to check that this description is correct in a particular model such as Rezk's $\Theta_n$-spaces.