In these notes the author defines a Lie algebroid as follows:
A Lie algebroid on $X$ is a sheaf of vector spaces $\mathcal{A}$ together with:
$(a)$ a structure of $\mathcal{O}_X$-module on $\mathcal{A}$;
$(b)$ a structure of $\mathbb C$-Lie algebra $[\cdot, \cdot]$ on $\mathcal{A}$;
$(c)$ an action of $\mathcal{A}$ on $\mathcal{O}_X$ by derivations given by an $\mathcal{O}_X$-linear map of $\mathbb C$-Lie algebras $\sigma:\mathcal{A}\longrightarrow \mathcal{T}_X$ called the anchor such that $$[a, f b] = \sigma(a)(f)b + f[a, b] .$$
What would be nice references which use the above approach to Lie algebroids?
I only found this exact definition in the notes of Paul Bressier on Lie algebbroids and characteristic classes, which you have mentioned. Another reference is the paper by Charles-Michel Marle; see the definition is $3.1.1$ on page $11$, which is slightly different, but has many more references in the bibliography.