Hi there. I think you're interpreting "average" as being equivalent to "mean", but this isn't my understanding of the term. Both the "median" and "mean" are measurements of "averageness". I use "median" when I'm talking directly about figures or a diagram. I use "average" when I'm making some general point, and only after I've already specified what my exact measure of central tendency was.
I've never heard anyone use "median" and "average" interchangeably who wasn't confused about what "median" and "mean" mean, and every math class I've ever taken involved a reminder that "median" and "average" are different.
Dig up arcane usages to support your confusing wording all you want, but the point of language is communication, and when you use words in significantly different ways from other people, you're communicating unclearly.
Then your maths professors were more lax than mine, who were always careful to distinguish between the general term "average" and specific terms like "mean" and "median".
As for the accusation that I'm being arcane - all I can do is point you to that most un-arcane and accessible of all reference works, Wikipedia. It spells it out quite clearly, and agrees with me entirely.
1. In general usage, when people say 'average' they almost always mean the mean, not the median.
2. Pointing out that something is on Wikipedia does not prove how un-arcane it is. I'm sure there are plenty of topics on Wikipedia that are only understood by a small group of people. Wikipedia doesn't require that all pages be common knowledge to the general population before being accepted.
People do often use the word "average" when they mean "mean" - but they also often use it when they mean "median". It depends on the context. For instance, when I say "the average family has an income of X" or "the average man is Y cm tall", the concept that most reasonably applies is the median, not the mean.
The simple fact is that "average" is a general term which is only used correctly once you've clarified what your exact measure is.
No, those clearly refer to the mean not the median. People do not ask the question "Given the range of heights in the population, what is the middle value", they want to know the mean height of the population.
This is especially easy to see when people talk about "average income", since it's clearly skewed by there being a long tail on the high-income side which pushes up the average salary. There's a reason why income is generally bracketed instead, and split into quintiles.
Look more carefully at my wording - most of the time, when you hear a news report using wording like "the income of the average family..." (as opposed to "average income"), you're hearing a median value, not a mean. The newsreader probably doesn't know it, but this is precisely because the median is less skewed by that long tail you mention.
http://en.wikipedia.org/wiki/Average