Triangulated categories were introduced in the 1960s by Grothendieck and Verdier in order to develop homological algebra in the framework of derived categories. An example of a triangulated category is the category $\mathbf K(\mathscr C)$ of complexes in an additive category $\mathscr C$ with morphisms the homotopy classes of morphisms of complexes.
In 1991, Neeman published a paper (“Some new axioms for triangulated categories”) proposing a variant of the definition which is slightly stricter: in the $\mathbf K(\mathscr C)$, the distinguished triangles are those $X\xrightarrow f Y \to C_f$ which are explicitly given by the construction of the cone $C_f$.
This definition does not seem to be the one adopted in Neeman's book (Triangulated categories, Princeton University Press, 2001).
Here is one interest of Neeman's definition: when a triangulated category $\mathscr T$ is endowed with a $t$-structure, one gets a canonical functor $\mathbf D(\mathscr C)\to \mathscr T$, where $\mathbf D(\mathscr C)$ is the derived category of the heart $\mathscr C$ of the given $t$-structure. In Verdier's framework, the existence functor lies in additional (possibly nonexistent) choices. An example of this situation is given by Beilinson's theorem (“On the derived category of perverse sheaves”, 1990) that on a scheme the derived category of perverse $\ell$-adic sheaves is equivalent to the constructible derived category.
I am on the verge of teaching a basic introductory course on perverse sheaves which will force me to give one definition. Which one would experts recommend that one chooses?
Edit: I had initially thought that the definition in Neeman's book was the one in his paper, but this does not seem to be the case.