It seems that there are different conventions in the literature as to what is a locally compact space (when the space is not supposed Hausdorff).

The two main non equivalent definitions I've seen are :

(LC1) every point has a compact neighborhood

(LC2) every neighborhood of any point contains a compact neighborhood of the point.

I am wondering if there is a reason as to why one might prefer one definition to another. More precisely, in practice, what definition is really useful (yields interesting results), or are they are both important in their own right ? In that case why not give a different name to those definitions ?

Naively, when one looks at the definition of locally connected space, one does not use the version (LCn1)"every point has a connected neighborhood", but always (LCn2) "every neighborhood of any point contains a connected neighborhood of the point". Is there a deep reason as to why LCn1 is never considered, but LC1 is ?

I am aware that for Hausdorff space, LC1 and LC2 are equivalent, and since LC1 is easier to check, one might prefer that as a definition, but this argument is unconvincing if LC2 is actually more useful for non-Hausdorff space.