You've excluded some of the definitions of average however. It's been taught at schools for decades in maths classes that "average" can refer to mean, median or mode. I took my definition from the same source as you, it does include the reference to median being a possible meaning.
If you counter-argument is that average
cannot mean median, under any circumstances, then it's not enough to show that mean is the "usual" meaning. "X cannot be Y" is a false statement if you can find even a single X which is Y. It doesn't matter if almost all X's are not Y, even a single counter-example disproves the statement.
Here's the Oxford dictionary definition:
https://en.oxforddictionaries.com/definition/average1) A number expressing the central or typical value in a set of data, in particular the mode, median, or (most commonly) the mean, which is calculated by dividing the sum of the values in the set by their number.
Here's the Merriam-Webster's Dictionary definition:
https://www.merriam-webster.com/dictionary/averageDefinition of average
1a) : a single value (such as a mean, mode, or median) that summarizes or represents the general significance of a set of unequal values
In each case, "median" is referred to in their primary definition. Here's more detail from wikipedia:
https://en.wikipedia.org/wiki/AverageFirst, you'll note that mean, median and mode are all listed in this page. This is the introduction
In colloquial language, an average is the sum of a list of numbers divided by the number of numbers in the list. In mathematics and statistics, this would be called the arithmetic mean. In statistics, mean, median, and mode are all known as measures of central tendency.
So in other words it's improper colloquial usage to use "average" to only refer to the mean, because that creates an ambiguity with the usage of the term in statistics, which is where the process originated.