median

[parisse]

Wikipedia (French) tells
"Pour une liste ordonnée de 2N éléments, toute valeur comprise entre l'élément N et l'élément N+1 est une médiane;"
(evey value between element N and N+1 is a median if the list has 2N elements).
Wikipedia (English) "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", observe the word "usually".
If you are using median to describe a statistical serie, using one definition or another is not important. Those who insist on the arithmetic mean definition are using medians for a different purpose, not related to statistics.