4 October, 1935
For the particular dates mentioned in the question:
- When Rowling gave that 2000 interview (the transcript dates it to 16 October), McGonagall would have recently turned 65
When the series ends (July of 1998 for the main story), she would be a few month shy of 63
- In the epilogue, 1 September 2017, she'd be just a month shy of 82
When we first see her, 31 October 1981 (the first chapter of Philosopher's Stone), she would have just turned 46
When Harry first meets her, 1 September 1991, she would have just turned 56
There's no convenient time where Rowling's statement is accurate, and no easy way to explain it away without a generous rounding error. I'm going to take a leaf out of Slytherincess' book and say that JKR is really really terrible at her math.
Okay, how do we know the date?
Her Pottermore character profile gives her date of birth as 4 October. This has been known at least since 2006, when Rowling posted birthday wishes on her website; although that aspect of her website has not been archived, the Lexicon reports on it.
From Order of the Phoenix, we know that she began teaching at Hogwarts in December 1956:
'How long have you been teaching at Hogwarts? Professor Umbridge asked.
'Thirty-nine years this December,'1 said Professor McGonagall brusquely, snapping her bag shut.
Order of the Phoenix Chapter 15: "The Hogwarts High Inquisitor"
From her biography on Pottermore2, we know she worked at the Ministry for two years before accepting a post at Hogwarts:
Finally, after two years at the Ministry, she was offered a prestigious promotion, yet found herself turning it down. She sent an owl to Hogwarts, asking whether she might be considered for a teaching post. The owl returned within hours, offering her a job in the Transfiguration department, under Head of Department, Albus Dumbledore.
So she started working at the Ministry some time in 1953 or 1954. We can do better by looking earlier in her biography; she began at the Ministry just after the summer following her Hogwarts graduation:
Upon graduation from Hogwarts, Minerva returned to the manse to enjoy one last summer with her family before setting out for London, where she had been offered a position at the Ministry of Magic (Department of Magical Law Enforcement).
So this is at most three months after graduation, most likely in late July or August. The year must have been 1954, otherwise she'd have gone from July/August of 1953 to December 1956, which is three years.
So we know that she graduated from Hogwarts in June 1954, so her seventh year spanned September 1953-June 1954. Working backwards, we can determine that her first year began in September 1947 and, because of her October birthday, she'd have been about to turn 12.
1 Order of the Phoenix is set from September 1995 to June 19963, and this conversation takes place some time between September and October4, so we go back 39 years to December 1956
2 Account no longer required (as of September 2015) to access Pottermore content.
3 Sanity check time. Per PearsonArtPhoto's answer to the linked question, Harry's second year ran from September 1992-June 1993. Thus Harry's third year would have been 1993-1994, his fourth 1994-1995, and his fifth 1995-1996. Sanity confirmed.
4 This conversation occurs in Chapter 15. The first Hogsmede weekend of term , which is said to be over the first weekend of October (and also includes the first meeting of Dumbledore's Army, in the Hog's Head pub), occurs in chapter 16