Exceptionally close, but still not quite there. You are right that the second number (in common parlance) denotes what kind of note gets one beat... except in compound time. 6/8 is very common, but that signature denotes "two beats per measure, dotted quarter gets one beat." As defined, the second number has to be a power of two, not just a multiple of two. It does get very complicated, though... 6/4 is six beats per measure, quarter gets one beat. And Bach often wrote in smaller divisions, 12/16 time is not uncommon for him.
Naughty, I am very familiar with Christ lag in Todesbanden, and many other chorale preludes and cantatas. Sorry you had such a traumatic experience. I'm guessing that when you and I were both going through our early music education, we weren't taught nearly enough about how to sightread.
But back to the Swan Song... I didn't really mean to make this a treatise on "Music notation: theory and practice." Sorry, Stronzo and others.