What are housekeeping and constitutive genes?
I think the best answer is both: they are continually transcribed at a rate that maintains the synthesized products at a (roughly) constant level. These terms are really just descriptive terms, there is no true biological dichotomy, it's just that some genes are expressed pretty much all the time and others are only produced when needed, for example genes for proteins needed at a particular stage of the cell cycle.

As @canadianer mentions in a comment, there are also other forms of post-transcriptional regulation, such that constitutive transcription doesn't necessarily mean constant production of protein at an equal rate.

Wikipedia has quite a list of different "housekeeping genes" and you will see the list is quite extensive. It certainly isn't true that all of these genes are expressed at exactly the same levels all the time, it is simply that their expression is consistent enough to be used as an approximate standard when comparing expression levels.

If I could produce a definition that reflects reality better than the ones you found, I would amend the Oxford definition to something like:


  Constitutive Genes are genes that are expressed following interaction between a promoter and RNA polymerase at a level that doesn't seem to vary much across cells of the same type, doesn't require a very special transcription factor, and so far biologists haven't found an experimental condition that greatly impacts the expression level.