Well, I think there is some evidence that the two are supposed to be proportional in certain cases. For example, look at p. 178 here where we have BWV 661a in common time with sixteenth notes versus the final version in cut time with eighth notes. However, there are plenty of other examples where Bach did not change the note duration but just the time signature (see a couple of pages above on p. 174, not to mention examples in countless of other works). I think that he, like any other human being, sometimes simply changed his mind and did not have the idea of a "fixed beat".
Regardless of whether or not cut time should keep the same "beat" as common time, however, I do think cut time means that one should be able to count each measure in two rather than in four, which provides some kind of a lower bound for the tempo. Some recordings of this work (especially older ones), feel a bit too slow to be in 2/2 rather than 4/4. (I think that the overture from the 4th keyboard Partita, BWV 828, which is also in cut time, suffers similarly, although tempi for that piece are often even slower.)