In the first text, there is a definition:
A derived stack is a derived Artin $n$-stack if it is an $m$-geometric derived stack for some $m$, and if $t_0(F)$ is an (Artin) $n$-stack. A derived Artin stack is an Artin $n$-stack for some $n$.
In the second text, there is a definition:
(1) A derived stack is a derived $n$-Artin stack if it is of the form $|X_∗|$ for some smooth groupoid of derived $(n − 1)$-Artin stacks $X_∗$.
(2) A morphism between derived $n$-Artin stacks $f:X\rightarrow Y$ is smooth if there exists smooth groupoid of derived $(n − 1)$-Artin stacks $X_∗$ and $Y_∗$, a morphism of groupoid objects $f_∗:X_*\rightarrow Y_∗$ with $f_0:X_0\rightarrow Y_0$ smooth, and such that $|f_∗|$ is equivalent to $f$.
A derived stack is a derived Artin stack if it is a derived $n$-Artin stack for some $n$.
In the first text there is also an admonishment:
The reader is warned that there is a small discrepancy for the indices in the notions of $n$-Artin stack and Artin $n$-stack. For example a scheme is always an Artin $0$-stack, but is only a $1$-Artin stack. It is a $0$-Artin stack if and only if its diagonal is an affine morphism...
Does the term "derived Artin stack" mean exactly the same thing in the first text and in the second text? If the answer is positive, please provide a detailed proof.
Is there any constraint on the two indices (e.g. given a derived Artin stack, minimal $n$ in the second definition is a bounded linearly in terms of minimal $n$ in the first definition)?