When someone says, “I’m 60 years old,” he means, “I have had my 60th birthday and have not yet had my 61st birthday.” Traditionally, someone born at any time from 00:00 (midnight) on june 1 to 23:59:59 (just before the next midnight) on June 1 has a birthday on June 1 every year. At least in the UK, by tradition, Independent of what time of day he was born, he will celebrate his birthday starting at 00:00 on each June 1, and will end his birthday at 23:59:59 on each June 1. 

Let’s set aside the effect of time zones on birthdays for another post. And let’s set aside the effect of changes of calendar, like the Julian-to-Gregorian change in 1752 in the UK, for another post too.

If the 60-year-old is speaking on a particular date , say yyyy/mm/dd, his 60th birthday could have been celebrated, at the latest, on yyyy/mm/dd. (So he’d be speaking on his 60th birthday). It could have been celebrated at the earliest on the day after (yyyy-1)/mm/dd. So his birthdate, the date written on his birth certificate, could have been anything from the day after (yyyy-61)/mm/dd up to (yyyy-60)/mm/dd.

“The day after” is just read off the calendar in the normal way. If (yyyy-1)/mm/dd is 1953/Dec/31, the day after is 1954/Jan/1. “The day after” 1953/Feb/28 was 1953/Mar/1. “the day after” 1952/Feb/28 was 1952/Feb/29, because 1952 was a leap year.

If you use the birthdate calculator to work out birthdate from someone’s stated age on a given day, and only fill in the ‘years’ field of the age, it’s understood that  you mean, they’ve celebrated that many birthdays, but you’re not sure when their birthday actually is. The birthdate calculator computes an earliest birthdate and a latest birthdate that could possibly make sense, according to the convention described above.