The questioner asks “What does ‘transcribed at a relatively constant level’ mean?” The answer is absolutely nothing!
A gene can be transcribed at a constant rate, or maintained at a constant level (in so far as ‘level’ — a word that should be avoided — is taken to mean ‘amount’). Perhaps it means ‘transcribed so as to maintain a relatively constant cellular concentration of the product’. It is unclear.
I’m afraid that it is a waste of time trying to find an authority for definitive meanings for terms that were coined in a period of incomplete scientific knowledge and have changed their meaning or importance as science has progressed and time has passed. Wikipedia is written by anyone who cares to contribute and its authority tends to vary with the amount of interest experts in the field have. For computing science you are on a winner, for molecular biology you aremay not be.
As far as the term housekeeping gene goes, a Google ngram shows that it appeared about 1980, at a time when little was known about the regulation of the expression of eukaryotic genes. The term expressed the simplistic idea of genes that were expressed continually (as in the examples quoted) as opposed to those induced by particular stimuli like sexual differentiation or perhaps change in diet. In fact little was known about inducable genes at the time, and the concept was very much based on an analogy with bacterial systems, which are different in many ways. It is still useful in a general sense, but no more.
PS
I'll editI've edited the Wikipedia entry when I have a moment. We'll see if anyone cares. (Or youI can doedit it yourself, but anyone else can switch it back to the original. In that case it is best discussed.)