Let's sum up.
The "min_stay" works as described below:
If any part of a new reservation includes the "min_stay" restriction date(s), it cannot include less than the minimum number of days on the stay restriction in order to be booked. For example, if a "min_stay" restriction is set up on Saturday for 3 nights, a guest trying to book a reservation for Friday through Saturday would not be able to book a reservation, as it includes fewer than the minimum number of days on the "min_stay" restriction. If a guest wanted to stay Friday, Saturday, and Sunday, then he would be able to book the reservation. Or another example: A three-night "min_stay" is set for August 16th. Any guest whose stay includes that day (August 16th) must stay at least 3 nights.
The "min_stay_arrival" works differently:
New reservations arriving on the date where a "min_stay_arrival" restriction is in place, must meet the minimum number of days on the "min_stay_arrival" restriction in order to be booked. For example, if you have a "min_stay_arrival" restriction on Saturday for 3 nights, a guest trying to book a new reservation for Saturday only, would not be able to book a reservation, as it does not satisfy the "min_stay_arrival" restriction of 3 nights. If the new reservation were arriving on Saturday for 3 nights, then the reservation would be able to be booked. A guest trying to book a reservation for Friday and Saturday would also be able to book the reservation, as it does not arrive on Saturday and is not subject to the "min_stay_arrival" restriction.