Yes, there are definitely two kinds of months. The crazy calendar months with 12 of them all having various days in them from 28 to 31, so it all "comes out right" and lunar months (the time system NATURE intended) that is every four weeks.)
I agree that your anniversary should be on the same date of the calendar month.
On that note I think that this is a fine example of how humans feel the need to complicate things. (by making up a calendar system that goes against natures calendar-the moon cycle is every 4 weeks, the menstrual cycle is every four weeks, a pregnancy is actually 10 lunar (four week) months starting from the woman's last period-not the close to 9 calendar months that we go by nowadays.
Going with nature is definitely much simpler.