I don't know who first coined this term, but they should be shot, IMO. This is a silly, non-sensical, incorrect term. There is no such thing as a half-cage. A cage is, by definintion, an enclosed space. You either have one or you don't. You cannot have half of one. Would you take small children to the zoo where the lions, tigers and bears were kept in half-cages? Would you live next to the prison where the inmates were kept in half-cells? Would you eat your soup out of a half-bowl? Would you keep your goldfish in a half-tank?
What you have is a four point roll BAR, nothing more, nothing less. If and when you add bars to the front, then you will have a roll CAGE.
I have a similar Autopower bar in my car, except I got mine with the bolt in harness bar and diagonal bar. In hindsight, I might have been better getting the welded verison like you have. It looks like you reinforced the mounting points at the floor. I did the same. I was very disappointed to see that the intended mounting points were right over the rubber drain plugs in the floor. Actually, in hindsight, I would have been better off having a custom bar built.