I just googled it and all the sites I looked at come up saying there are 3 types of average - mean, mode and median
I did statistics at university, in a different country and long ago, and I'd never come across the idea of a median being "a type of average" until Mumsnet alerted me to the fact that this is what British schoolchildren are taught.
I contend that no-one who knows the difference between a median and a mean (of any kind) would ever use the word "average" to describe a number they knew to be a median.
99% of the time time, "average" means "arithmetic mean", and 99% of the rest of the time it will mean "geometric mean". (Those statistics freshly sucked out of my thumb, in case that's not obvious.)
To be fair, I think in my first ever statistics lecture, a phrase something like "measures of the centre" was used to encompass all the measures that many of you like to claim are covered by "average", and that is relatively a bit of a mouthful.