Steve,
We are in deep details here (luckily far from being deep in something else), where things become a bit woolly and pleasantly challenging, especially since the number of terms is smaller than that of the items they decribe, especially when uses are taken into account.
Before suggesting anything, I had a look in the OED to make sure I was remembering what I thought I was remembering, and from that I would say that a rebate is a through thing, from end to end of the edge, and that a mortise can be there without a tenon, serving a rather wide range different purposes, the combined term only appearing in the second sense in the OED.
By the way, the word rebate is consistently listed in dictionaries with reebait mentioned before rabbit, as some of us (still, maybe sillily so) prefer it; and in my other favourite dictionary, which is very distinct in its distinction between BE and AE, rabbet is listen as another word for rebate.
But I believe usage is the key, and doubtlessly usage differs, with time, country, phase of the moon, and other important influences.
In any case, the visitors will doubtlessly be doubtless.
