Your Adorama monolight is designed for modifiers using a Bowens S mount, so the Bowens stripbox should work just fine. The Bowens mount has 3 tabs on the modifier that you twist to lock into place on the light itself. It'a very solid fit and the modifier won't move once it's locked in place.

Paul C Buff uses a different mount (Balcar). This is a grooved ring on the modifier, and there are four spring-tensioned clips on the light that grip this groove and hold the modifier to the light. I've used this with Einsteins and the advantage is that you can angle the modifier any way you like vs. being fixed in one of 3 positions with Bowens.

To use the Buff softbox with your Bowens light, you will have to buy a new Bowens mounting ring ($10-$15), take the Balcar ring off the Buff softbox, and then fix the Bowens ring to the softbox. It's not that difficult -- Buff lights use thumb screws to hold the ring onto the softbox, so you don't even need a screwdriver -- but remember you are doing this with a rented softbox, so be careful not to mess it up. And don't forget to convert it back before you return the softbox.