The date shown in the database is always going to be different than the date shown in the UI unless you're in the UTC timezone. The database dates are all UTC.
You mentioned auto-debit runs at 9am MST, which is -7 UTC, so invoices auto-debited would be closed at 16:00 UTC.
The dates you mentioned are.
Bill Date: February 8, 2016, 7am MST
Date Due: February 15, 2016, 11pm MST
Date Closed: February 15, 2016, 9am MST
Bill Date: February 10, 2016, midnight MST
Date Due: February 17, 2016, midnight MST
Date Closed: February 16, 2016, 9am MST
The time of day invoices are due don't have much of a bearing on the auto-debit since it looks at the entire day to find invoices. It could be throwing things off if the timezone conversion comparison is incorrect. If you look at more examples, do you see consistent behavior for invoices whose due date time is between 00:00:00 and 06:59:59 (in the database) being auto-debited the same day, and otherwise auto-debited the day before if the time is 07:00:00 or greater?