The type of month described is easily seen as a month other than February that begins on a Saturday.

The major gist of the puzzle is which year was the first one of the current millennium. Every millennium since the first has begun on a year ending in the digit 1, so it was 2001 (where we got the fall of the twin towers instead of a space odyssey).

A quick look at any perpetual calendar or app shows that September and December of that year met the criterion. The first Tuesday of that first such month, in DD/MM/YYYY format, was 04/09/2001. It was the second Tuesday that was the killer.