Local maxima 

Feller says that unimodal distribution is a distribution function that is convex up to the mode and concave beyond it. Thus it may have density taking maximal values on an interval (say uniform distribution on [0,1]). In this case all points in [0,1] are local maxima (or there aren't any, if we consider strict 'peaks' only). Therefore I suggest a more general definition of a unimodal function: a function whose local maxima make up an interval.

Moreover, it might be worth replacing '(this allows for the possibility of a non-zero probability for x=m)' with '(in this case the distribution may have density everywhere except, possibly, m; this allows for the possibility of a non-zero probability for x=m)'