這兩種講法是否都正確？

Both are correct.
In (1) half a mile = half of a mile.
In (2) a half mile = a half-mile. "half mile" or "half-mile" is a compound noun meaning a distance unit of half a mile". if "I run a mile every day" is correct, so is "I run a half mile every day", as both "mile" and "half mile" are measurement units of distance.

但書本說當half 作名詞用時，可以這樣說：
a half hour 半小時
a half pound 半磅
a half mile 半英里

你同意嗎？

the first statement is correct, but not the second one .

We walk a mile every day.
We walk half a mile every day.

We walk one mile every day.
We walk half mile every day.

We walk a distance of one mile every day.
We walk a distance of half mile every day.