I have never seen median referred to as a "type of average" in any context.

Is there any foundation to this?

Quite literally the first result in Google: http://www.dummies.com/education/math/pre-algebra/the-three-...

Also, from webster's:

"av·er·age\ˈa-v(ə-)rij\ noun

: a number that is calculated by adding quantities together and then dividing the total by the number of quantities

: a level that is typical of a group, class, or series : a middle point between extremes

Full Definition

1 a : a single value (as a mean, mode, or median) that summarizes or represents the general significance of a set of unequal values b : mean 1b 2 a : an estimation of or approximation to an arithmetic mean (...)"

It's common to see arithmetic mean and average to be treated as synonyms - but the mean is really just a type of average.

You might like this blog post from John Myles White: http://www.johnmyleswhite.com/notebook/2013/03/22/modes-medi...

It compares mode, median, and mean as each being specific implementations of a more general concept of discrepancy.

Colloquially, median and mode are used as average more than you'd expect. For example, we might say "the average joe is right handed", which is most likely the mode.