Actually, mean, median and mode are measures of central tendency not "3 kinds of averages". Also, median is not "halfway point between the highest number of points owned and the lowest." You are referring to the midpoint of the range. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking the middle one. If there is an even number of observations, then there is no single middle value; the median is then usually defined to be the mean of the two middle values. While measures of central tendency are important, the measure of dispersion is as important to understand the spread of the observations.
In the end I feel that the DVC member used "sloppy statistical" jargon. The same way I hear people interchange statistic and parameter, "mu" and "x bar", or use sigma when they are actually talking about sample standard deviation (s). I will get off my soapbox now.