I am trying to see some standard examples of Lie $2$-algebroids. The first entry in Google search takes me to Madeleine Jotz Lean's work Lie 2-algebroids and matched pairs of 2-representations — a geometric approach. There is a mention of the term "split Lie $2$-algebroid" and "Lie $2$-algebroid". It looks like these two are the same concepts. For confirmation, I backtracked to the work Higher Extensions of Lie Algebroids of Yunhe Sheng and Chenchang Zhu where the notion of split Lie $2$-algebroid (more generally split Lie $n$-algebroid) is introduced.
A split Lie $n$-algeroid consists of a non-positively graded vector bundle $\mathcal{E}=E_0\oplus E_{-1}\oplus E_{-2}\oplus\dotsb\oplus E_{-n+1}$ along with a morphism of vector bundles $\rho:E_0\rightarrow TM$, a collection of maps $\{l_i:\Gamma(\Lambda^i \mathcal{E})\rightarrow \Gamma(\mathcal{E})$ satisfying certain conditions. Please see Definition 2.1 in Higher Extensions of Lie Algebroids for a precise definition.
This looks exactly the same as the notion of Lie $\infty$-algebroid where there is a bound on the grading part of the vector bundle.
For me, the prefix "split" means there is some choice made, as in splitting of an exact sequence of vector bundles.
Are the notions Lie $2$-algebroid and split Lie $2$-algebroid really different?
