Right, good news, possibly on more than one count!
I programmed the LSI1024 tonight and it programmed in one go which is better than mine did when I programmed it. I had to go over mine 3 times before it would fully program. One thing though which I had completely forgotten. Even when I powered the LSI from the programmer the software still reckoned VCC was at 0v. Curious since I'm sure I didn't have any wiring wrong. I just disabled the voltage check and it programmed / verified just fine.
Point is though that those Chinese programmers you have could still be OK yet. If there is any option in the software to disable the check for chip presence or chip voltage, go ahead and try selecting that. You never know it may go ahead and program it anyway. I don't believe you can damage the chip. Don't take my word alone for that but I believe the worst that could happen is you end up with a chip that is only partly programmed...