On the definition of locally compact for non-Hausdorff spaces

To me, the second definition of local compactness is much to be preferred for the simple reason that such locally compact spaces $X$ are exponentiable in $Top$, meaning that $X \times -: Top \to Top$ has a right adjoint $(-)^X: Top \to Top$ (even without the Hausdorff condition), and all this implies (such as $X \times -$ preserving coequalizers). In fact the necessary and sufficient condition for exponentiability, called core compactness, is only a mild generalization of local compactness (and equivalent to it under the Hausdorff assumption).

The notion LCn1 just boils down to "the connectedness components of the space are clopen". If this property does indeed show up somewhere, I would expect that the latter is a more convenient way of expressing it.

LC1 on the other hand does indeed seem to capture some intuition about "this space has some properties of compact spaces, but might be too large to be compact". Naming it locally compact is probably a mistake coming from the equivalence to actual local compactness in Hausdorff spaces.