After reading the "der weg zum kunstlerischen klavierspiel" by josef dichler,in an example it has that trill from bethoveen opus 111 he gives 3 fingerings
51,42,51,42
41,52,41,52
42,51,42,51
and says on a side note "that the first fingering (51,42,51,42) comes from bethoveen and can just be executed with difficulty (and he may thinks that it is a mistake) but at another side note it says "Maybe bethoveen means that the trill should start with the neighbour notes (that is Bb,and Eb) then the fingering of bethoveen would the same as the last fingering (that is that 42,51,42,51 becomes then 51,42,51,42 and so on.)
He also adds that the second fingering is not good for fast trill though it feels normal.and adds that we should use the last trill,42,51,42,51
Hope that helps