I, myself, am revolted with such a definition, too.

It seems, according to Wikipedia, that there is no consensus on the definition. And probably, few people care about it, because in Hausdorff spaces, all the definitions are equivalent. In addition to the definition you present, one could also say that a space is locally compact when every point has a closed compact neighbohood. In general, even if a neighborhood is compact, it does not mean its closure will be compact as well.

In the Hausdorff case, all closed subsets of a locally compact space are compact, since every compact set is closed. In the Hausdorff case, it is also true that if $K$ is a compact neighborhood of $x$, then $x$ has a neighborhood filter base made out of compact sets. This is because $K$ is a normal space.

I, myself, am revolted with such a definition, too.

It seems, according to Wikipedia, that there is no consensus on the definition. And probably, few people care about it, because in Hausdorff spaces, all the definitions are equivalent. In addition to the definition you present, one could also say that a space is locally compact when every point has a closed compact neighbohood. In general, even if a neighborhood is compact, it does not mean its closure will be compact as well.

In the Hausdorff case, the closure of subsets of compact sets are compact, since every compact set is closed in this case. Also, in the Hausdorff case it is true that if $K$ is a compact neighborhood of $x$, then $x$ has a neighborhood filter base made out of compact sets. This is because $K$ is a normal space.