Yes, in essence what they are offering you is a true drop-shipping service, that actually even goes a little further than a regular blind shipping service. It sounds like this supplier not only understands drop-shipping but has been seriously considering it, just hasn't quite put the processes into practice yet.
It could be that they see an opportunity to use you to test it out to see how it works for them, that's, of course, a good thing for you if you want to deal with them.
The only cause for concern that I can see is that given there is no drop-ship account option for you when processing the order, you will need to make sure that the supplier knows the order is for one of your customers and not just someone ordering off the web themselves. That, of course, would see you miss out on the commission as well as see the package go out branded.
My guess is that they may well be going to provide you with a coupon code to enter at the checkout which will identify your purchase as yours, despite what the shipping address is. That, of course, may also be done by the billing address being different as well, so a couple of ways they can do it from their back-end, just make sure you clarify how that works to protect yourself.
So no need to be too worried about what they want to call it, if they're happy to provide you with the service you require, then who cares about the name :)