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 is too large to be compact". Naming it **locally compact** is probably a mistake coming from the equivalence to actual local compactness in Hausdorff spaces.