My opinion, and that of many other people although not of everyone, is that the "correct" notion is that of stable ∞-category.

Now, this is not a category in the strictest sense, rather a generalization of the notion of category known as an (∞,1)-category, or ∞-category for short, where to any pair of objects $x,y$ there is an associated homotopy type $\mathrm{Map}_{\mathcal{C}}(x,y)$, usually called the *mapping space*. You can get a category from that datum by taking the connected components $\pi_0\mathrm{Map}_{\mathcal{C}}(x,y)=:[x,y]$. The resulting category is called the *homotopy category* $h\mathcal{C}$, and can be seen as the best approximation you can give of an ∞-category using an ordinary category.

You can talk about limits and colimits in an ∞-category, and in fact pretty much all of classical category theory goes through in this more general setting without problems (although with the occasional very important modification). Then you can say that an ∞-category $\mathcal{C}$ is *stable* if it satisfies the two following conditions:

It has a zero object (i.e. an object $0$ such that $\mathrm{Map}(x,0)$ and $\mathrm{Map}(0,x)$ are contractible for every $x\in\mathcal{C}$).

It has all pullbacks and pushouts and a square (i.e. a diagram of the form $[1]\times [1]\to\mathcal{C}$) is cartesian iff it is cocartesian.

As you can see, it is a fairly simple definition. It can be rephrased in a few equivalent ways, some of which are rather easy to check. This notion has a few very important properties:

For every stable ∞-category $\mathcal{C}$, the homotopy category $h\mathcal{C}$ has a canonical triangulated structure.

All triangulated categories that actually show up in mathematical practice usually come equipped with a specific stable enrichment (i.e. a stable ∞-category whose homotopy category is the triangulated category you were thinking about). In a few cases, the stable ∞-category is actually easier to define.

There are examples of triangulated categories that do not come from a stable ∞-category. All the examples tend to look unnatural, and we would very much like a definition that excludes them.

In stable ∞-categories, a lot of the theorems that one would expect to be naively true for triangulated categories are actually true. For example, cones are functorial, and you can define the algebraic K-theory of a stable ∞-category (while you cannot do so for a triangulated category!), obtaining the expected results (e.g. the algebraic K-theory of the stable ∞-category of perfect complexes over a ring is exactly the algebraic K-theory of the ring).

More abstractly, stable ∞-categories work well in families. For example, the functor sending a scheme $X$ to the stable ∞-category of perfect complexes over $X$ is a fppf sheaf (for an appropriate notion of sheaf of ∞-categories). This is *not* true for the corresponding triangulated categories!