Say the 24th is a Monday and you say that you’ll be doing something the weekend of the 24th, meaning the 22nd and 23rd. Isn’t that incorrect? I would say the weekend of the 24th means the 29th and 30th.
I'd argue that in the given month, the 'weekend of the 24th' is a misnomer; Monday is not normally a weekend day at all. Friday evening (the 21st of the given month) might just be counted as part of the weekend. And if it is a holiday weekend, then Monday might scrape as part of the long weekend, but normally, you would only reference a date that is part of the weekend.
Personally, I would not talk about the weekend of the 24th:
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I would say the weekend of the 22nd or 23rd.
Also the weekend of the 29th or 30th.
The week of the 24th has weekend days of the 23rd and 29th, those are the days that are on the ends of the week that contains the date of the 24th.
Specificity is best practice. As a speaker and writer of English, I'd say and write "weekend of Saturday (or Sunday) the __" to be as specific as possible.
However, I'm also a person who hears and reads a good deal of English. I'd assume that "the weekend of Monday the __" means the one in 5-6 days, and so on with other days of the week, unless it's obvious otherwise. I consider it likely that you'll run into the ambiguous usages, so take care to be on the same page.
It's a contextual (and therefore ambiguous) expression if you give a date outside the weekend. Barring even different calendar schemes (Chinese, Arabic, and Jewish Lunar calendars, for example), it's poor, but common, practice.
A week begins on a Monday and ends on a Sunday. That is why Sat and Sun are collectively known as the "weekEND".
So, for a week beginning on the 24th of a month:
(1) 24th would be a Monday. The 29th and the 30th would be the "weekend".
(2) The dates from 24 through 30 comprise the "week of the 24th". Note though that one could ALSO say "week of the 25th" or "week of the 30th" to refer to this week as long as the date used falls between 24 and 30.
(3) The dates 29 and 30 could be referred to weekend of the 29th or weekend of the 30th.