My 30 year old back rack has a tab that hangs down with a couple of holes. My two year old rack does not, but it has holes in the horizontal "plate". The first light had a plastic bar that accommodated this and I bolted it to the rear underside of the rack. After a few months that light worked its way loose, fell off and broke. The second light only had a plastic "loop" for a bar mount. I drilled a hole in that and used it to attach it to the underside of the plate.

The plastic quick-release just broke last week. So I drilled a small hole in the back of the light housing and used a small brass "mending plate" from the hardware store to bolt to the back of the light and then to the hole in the plastic loop.

If you shop for lights in person you may find one that will attach to your particular rack. There are a myriad of mounting option out there including rubbery straps and zip ties.

Otherwise, you may need to get creative at the hardware store using small plastic or metal parts. I find those challenges fun, especially if they involve less than $10. Others may not enjoy that and would rather have something that works "right out of the box".

As far as what to call the rack on the back of the bike, I have always called it a "back" rack, rather than a "bike" rack. I suppose for clarity it could be called a "rear package shelf". To further add to the confusion, the thing that you lock your bike to at a store or office is also called a "bike" rack. For clarity that could be referred to as a "parking" rack.