A day is defined as the hours between 00:00:00Z and 23:59:59Z (ie- UTC). Consecutive days are between those hours on immediate subsequent calendar days.
If Visit 1 happens at 13:24Z on Aug 31 and Visit 2 happens at 8:41Z on Sep 1 then you have visits on 2 consecutive days.
If Visit 1 happens at 23:50Z on Aug 31 and Visit 2 happens at 00:35 on Sep 1 then you have visits on 2 consecutive days.
If Visit 1 happens at 13:24Z on Aug 31 and Visit 2 happens at 8:41Z on Sep 2 (with no visits between 00:00:00-23:59:59Z on Sep 1) then you do not have visits on 2 consecutive days and the "visits on consecutive days count" for badges is reset.