There can be a big difference between when a Bar Mitzvah was actually celebrated, and the correct date according to Halacha.
There is also a difference between a bar mitzvah celebration (party), and the day when the boy was (or, should have been) called up to the Torah.
And, as someone pointed out, it would depend upon the level of observance of the particular family.
I am replying as the daughter of a teacher who prepared many boys for their bar mitzvahs. (I have not looked at the Chabad website, so some of what I am about to write might be a duplication.).
The secular date of birth was given as August 17, 1948. One piece of information that would have been necessary in order to find the corresponding Hebrew date was the time of day that the boy was born. Using the website hebcal.com, mentioned by others, the Hebrew date is the 12th of Av if he was born before sunset, and the 13th of Av if he was born after sunset, as Hebrew "days" run from sunset to sunset, not midnight to midnight. The Hebrew year was 5708. Thirteen years later was 1961, or 5721 in the Hebrew calendar. Again using hebcal, it can be seen that the 12th of Av, 5721 was July 25th, 1961, and the 13th of Av was July 26th. These were a Tuesday and Wednesday. The bar mitzvah day is not the "closest" Sabbath, but the first Sabbath after the 13th birthday. That Sabbath was July 29th. This is the simple answer to the question posed.
I would like to add a few points. I remember learning that it was important to make sure that 13 complete years had passed, so that the boy could be counted as part of a Minyan (quorum of 10 necessary for communal prayer) and thus could be called up to the Torah. Therefore, if his 13th Hebrew birthday happened to fall on a Sabbath, his bar mitzvah would be a week later, on the following Sabbath. I also want to point out that, even though most bar mitzvahs are celebrated on the Sabbath, a bar mitzvah can be celebrated on any day when the Torah is read. This occurs weekly on Mondays and Thursdays, and also on a number of holidays, including "Rosh Chodesh", the start of a new Hebrew month. One need only go to the Kotel (western wall) in Jerusalem on any Monday or Thursday to see many simultaneous bar mitzvah celebrations.
Brooklyn, New York, USA