Within the world of a college statistics textbook, the word "average" is an unofficial term of central tendency. Any term of central tendency (mode, median and mean) can be represented by using the word "average". The above example would be better represented with mean vs. median. The word "average" can be interpreted in many different ways.
Well, Marriam-Webster says:
average is the quotient obtained by dividing the sum total of a set of figures by the number of figures.
I'm ESL, though, and in my native language there is no word that means "mean, but maybe median or something like that", just a word for "mean" as defined above, so that might be coloring my understanding, too.
As a practical matter, if you say "the average house is $670k", would you interpret that to mean the mean or that $670k (a fairly precise number) is "some sort of central tendency" for the prices. To me, it's clear that the number $670k has been calculated in some way, and if it's the median they should say the median. (Although it's true that maybe the OP made an imprecise quote.)
My point still stands, you don't have to buy the median house either. In fact, half the houses are cheaper than that. ;-)