Without the Hausdorff hypothesis, some of these results break down with weaker notions of locally compact. Every closed set in a weakly locally compact space (= condition (1) in the definitions above) is weakly locally compact. But not every open set in a weakly locally compact space is weakly locally compact. For example, the one-point compactification of the rational numbers is compact, and hence weakly locally compact. But it contains as an open set which is not weakly locally compact.
without the hausdorff hypothesis some of these results break down with weaker notions of locally compact every closed set in a weakly locally compact space (= condition (1) in the definitions above) is weakly locally compact but not every open set in a weakly locally compact space is weakly locally compact for example the one-point compactification of the rational numbers is compact and hence weakly locally compact but it contains as an open set which is not weakly locally compact