That's everything you would typically see installed in the frame like that. That's just the cup and bearing. The more important part that would be on the fork steer tube (and would probably fall off easily after pulling the fork) is referred to here as the top cover assembly. It's similar to the crown race, kind of wedge like but it's for the upper part so it's like upside down.

Friend unfortunately from looking at it, I can't see all that well, however it looks like you might have an integrated cup headset that could use an unusual cup standard. You may not be able to replace the cups. Usually you can remove them with just a long screwdriver and a hammer. However the good news is that different parts might still be compatible or at least work pretty well together so you can probably just take the whatever parts are missing from the headset that you ordered. Normally headsets are "external cup" or what Google says it's ec34.

Integrated cup systems are outside of my skill set. It's not so much that I wouldn't be able to install one but that it's a little hard to tell exactly what standard you have from the pictures and what needs to be fixed. I could be wrong though, maybe you have external cup, I could tell from just a side view. I feel your pain buddy, incomplete headset systems can be a real pain in the ass.

Yeah to be clear, a lot of the 1 1/8" steerer systems have compatible parts regardless. You don't have 1" or 1.5" tapered or anything weird like that (at least it's very unlikely you have tapered)

That's it, yes it's attached to the old fork, it needs to be removed with a screwdriver and hammer and set into place on your new fork with a tube and a hammer. Sometimes you can use a PVC pipe or a seatpost but more likely you need to use a professional crown race setting tool to install it. Sometimes they're tight, sometimes not.