Any set of notes can be notated in any key signature that the person writing it down chooses. 

I don't have any huge interest in learning STJ myself, but since the matter of key signatures and modes reared their ugly heads, I had to read through the posts. LOL

I have found version with the same notes and chords notated as either G/Em with the C# notated as an accidental, or notated as D/Bm without C as an accidental. 

Both are technically correct.  There is no error in calling it either "key" or notating it as one sees fit, really.  I personally would usually tend to prefer to notate it in a key signature where I need no (or the minimum possible number of) accidentals.  But that is strictly a matter of personal preference and the decision is always at the discretion of whoever is writing it down.

But, for example, if a player is more familiar with G/Em, they may prefer to have it notated in that and have the accidental.  If the accompaniment begins on an Em chord, that may simply feel more intuitive to them.  Nothing wrong with that.   

If a score can be played to deliver the needed pitches, then it is "correct".  What key signature the person writing it down prefers to notate it in is their choice.  So both choices that have been mentioned in this discussion can be perfectly correct.  No need for argument, gentlemen.