Your Adorama Monolight was designed for modifiers with a Bowens S-mount, so the Bowens Stripbox should work just fine. The Bowens bracket has 3 tabs on the modifier that you turn to lock it to the light itself. It's a very solid fit and the modifier does not move once it's locked in place.
Paul C Buff uses another mount (Balcar). This is a grooved ring on the modifier, and there are four spring-biased clips on the light that grab this groove and hold the modifier to the light. I've used this with Einstein's and the advantage is that you can rotate the modifier as you like, instead of fixing it with Bowens in one of three positions.
To use the buff softbox with your Bowens light, you must purchase a new Bowens mounting ring ($ 10 to $ 15), remove the balcar ring from the buff softbox, and then attach the bowens ring to the softbox , It's not that difficult – Buff Lights use thumbscrews to hold the ring to the softbox so you do not even need a screwdriver – but remember, you do that with a rented softbox. So be careful not to mess it up. And do not forget to convert it back before returning the softbox.