One can say "in a month's time" and mean any number of things.
When one states "in a calendar month," however, one specifically refers to the period of time from the beginning of the month until the end of the month. The period of time varies with the month; if it's February, excepting Leap Year, then it's a 28 day period. If it's November, it's a 30 day period from the beginning of November until the end of that month.
A calendar month is not from the middle of one month into the next (although you're right; you'll find contradicting legal definitions saying otherwise, and saying for). For example, a calendar month is not from the eighth of July until the 7th of August; that period comprises two separate partial calendar months. It may cover a month's duration, but not a calendar month. A calendar month could be the month of July, beginning to end, or the month of August, beginning to end. Or one could specify a 30 day period. When the term "calendar month" is used, the use of "calendar" with the word "month" specifies that the month is defined by one of the 12 designated months of the year, and the month in question takes place from the designated beginning of that month, until the designated end.
If I have a medical certificate which was obtained on the 5th of january, for example, and expires six calendar months after the month in which it was issued, for example, the six calendar months begin after the month of issuance. If it was issued in january, my six months begin in February, and comprise Februay, March, April, May, June, and July. My certificate expires at the end of July, at the end of the 6th calendar month after the month in which it was issued.
However, even within the US legal system you'll find both definitions as accepted in different cases, you can use a legal dictionary to find this stuff.
Not just in the US legal system. For example, in McCombie v. Queen, CanLII - 2000 CanLII 354 (T.C.C.)
, the Canadian court kicks the idea around citing several other decisions and definitions in context, then finally concludes:
I cannot accept the proposition that a portion of a month is a calendar month.