My textbook sets this question:

In each of the following sentences, a word has been used in sentences in different ways. Choose the option corresponding to the sentence in which the usage of the word is incorrect or inappropriate.

The average of 3, 4, 5, 7. and 10 is 6 whereas the mean is 5.

The books says that it is wrong and median should be used instead:

In this sentence, mean has to be replaced with median for the sentence to make sense.

My question is: isn't mean correct? For years I've studying the word mean in mathematics, and now the book says: use median.

I don't think the sentence is wrong due to the mean being written as "6", because even if I write the wrong mean, the usage of the word is correct, I'm just writing the wrong answer.

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If you look at this link, it says, mean should be used for symmetrical data sets, and median for skewed data, In the above sentence, the data being skewed my guess is that is probably the reason median is being used and not that mean is written as 6 whereas actually median is 6.
Mean or Median

It is wrong to say "the mean of this set of numbers is 5". Not grammatically wrong, but mathematically wrong. The sentence could be made correct by changing it to, "the mean is 5.8", or you could change it to, "the median is 5". If the writer's intent was to tell you the median, then the error is saying "mean" when he means "median". If the writer's intent was to tell you the mean, then the error is saying "5" when the correct value is "5.8". As the sentence also mentions an "average", which is normally understood to be the same as "mean", it seems likely the intent was for the last number to be the median.

You seem to be getting stuck on the idea that the sentence could be corrected other than by changing "mean" to "median". Well, sure. Any time someone says, "How would you correct this sentence?", there are likely thousands of possible changes. Suppose someone said, "Thomas Jefferson was the first president of the United States". I could correct that by replacing "Thomas Jefferson" with "George Washington". I could replace "first" with "second". I could insert "Democratic-Republican" between "first" and "president". I could replace "Thomas Jefferson" with "Yuri Gagarin" and "president of the United States" with "human to travel in space". Etc etc.

Median is the middle value of a list of order numbers. In 10, 45, 600, 601, 615, 777, 999 the median is 601. The mean is the sum of the numbers divided buy the count of the numbers, for the above example that is 521.

Mean, median and mode (most popular number) are all types of average.

This is how it was taught to me, albeit long ago and with many more examples.

Your book asks if mean is the correct word when presenting the median number of your list. It then tells you that mean is wrong and median is correct, so that is fine.