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

 1. Gregory Ginot's [paper][1] defines a Lie $2$-groupoid to be a double Lie groupoid in the sense of the [paper][2] satisfying certain conditions.
 2. Rajan Amit Mehta and Xiang Tang's [paper][3] says that "The simplicial approach to $n$-groupoids goesback to Duskin in the discrete case, and the smooth analogue appeared in Andre Henriques [paper][4]. They define Lie $2$-groupoid to be a simplicial manifold $X=(X_k)$ with some condition on the horn maps.
 3. Matial del Hoyo and Davide Stefani's [paper][5] defines a Lie $2$-groupoid to be a Lie $2$-category and a $2$-category satisfying certain conditions.
 4. n-lab [page][6] on Lie $2$-groupoid 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][7] which do not contain much details.

Questions :

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

 


  [1]: https://webusers.imj-prg.fr/~gregory.ginot/papers/2-bundles.pdf
  [2]: https://www.sciencedirect.com/science/article/pii/0022404992901456
  [3]: https://arxiv.org/pdf/1012.4103.pdf
  [4]: https://arxiv.org/abs/math/0603563
  [5]: https://arxiv.org/pdf/1706.07152.pdf
  [6]: https://ncatlab.org/nlab/show/Lie+2-groupoid
  [7]: https://ncatlab.org/nlab/show/Lie+n-groupoid