I don’t think you’ll find a single dictionary or other reference work that defines “middle age” as including the early 30s. Wikipedia lists sources placing the start of middle age from 35 to 45, and the end anywhere from 54 to 65. It also describes middle age as being “around the third quarter of the average life span of human beings.”
One reason why I think of 30 as the beginning of middle age is that I recall a slogan of my youth: Never Trust Anyone Over 30. The implication was that 30 was the dividing line between the hip young folks and the uptight (and therefore untrustworthy) old folks.
I think there is a gap between young (under 30) and middle-aged. People in there 30s are not usually quite as into the latest trends and fads the way college kids and their peers are, but that doesn’t mean we’re stodgy old fogeys. Now that I’m 30, I admit I’m no longer young and hip, but I don’t expect to be middle-aged for at least ten more years. I’m just an adult.