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.

[Wikipedia]: http://en.wikipedia.org/wiki/Locally_compact_space
  (Article on Wikipedia about locally compactness)