The term Lie $2$-groupoid is used in the literature in more than one context. Some examples are given below:

1. Ginot and Stiénon's paper $G$-gerbes, principal $2$-group bundles and characteristic classes defines a Lie $2$-groupoid to be a double Lie groupoid in the sense of Brown - Determination of a double Lie groupoid by its core diagram satisfying certain conditions.
2. Rajan Amit Mehta and Xiang Tang's paper From double Lie groupoids to local Lie $2$-groupoids says that "The simplicial approach to $n$-groupoids goes back to Duskin [Higher-dimensional torsors and the cohomology of topoi: the abelian theory] in the discrete case, and the smooth analogue appeared in [Henriques - Integrating L-infinity algebras]." They define Lie $2$-groupoid to be a simplicial manifold $X=(X_k)$ with some condition on the horn maps.
3. Matias del Hoyo and Davide Stefani's paper The general linear $2$-groupoid defines a Lie $2$-groupoid to be a Lie $2$-category and a $2$-category satisfying certain conditions.
4. The n-lab page on Lie $2$-groupoids says "a Lie 2-groupoid is a 2-truncated $\infty$-Lie groupoid". No further discussion is given there. Clicking the link $\infty$-Lie groupoid takes you to the page Lie $n$-groupoid, which does not contain many details.

Questions :

1. Are these notions genuinely different notions introduced for different purposes or are they all the same candidate wearing different outfits?
2. Are there other notions of Lie $2$-groupoids in the literature?