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.
The simple fact is that "average" is a general term which is only used correctly once you've clarified what your exact measure is.