If I understand what you mean correctly, you wish to have a way to say DateTime.NextSunday? If that's the case, as far as I know there isn't a standard you can use for that.
The two examples you gave above are specific examples of a date you need. You would still express the date time using the ISO standard, you just have to prgrammatically define what each means.
How I would do this is in my UI, I'd have a button called "Next Week". Here in Canada, the first day of the week is Sunday, but that's not the same for everyone. Thus I'd write my code like this (c# is my language of choice):
DateTime today = DateTime.Now;
int daysToSunday = (DayOfWeek.Sunday - DateTime.Now.DayOfWeek) + 7;
DateTime weekStart = today.Date.AddDays(daysToSunday);
Note, I didn't test this, just a quick write-up.
Moral though, is that for a concept of time like those you've stated above, you need to code a solution to arrive at those dates.