How can zero be the median?
Talking about the median in this context actually means (the way I see it) comparing multiple sequences of coin flipping. If, in each of five sequences, the number of times the zero-difference point is crossed are (0,0,0,2,5), the median is zero. With six sequences (0,0,0,0,3,4), the median is also zero.
Zero can certainly be the middle number of an ordered set with an odd number of members or the average of the two middle numbers of an ordered set with an even number of members.
Basically, claiming a zero median is claiming that more than half the time, a sequence will NOT cross the zero-difference line.