Some things that come to mind, based strictly on the sound:
1) Starter is bad. If you got a reman unit from the auto parts store, take it back and get a NEW one.
2) Damaged flywheel
Now comes the silly question: Did you put the right bolts back in the right spot? Sometimes one bolt is longer than the other and if it's in the wrong place it will cause all kinds of problems. I've not dropped a Z starter recently, but I have seen imports with two different lengths of bolts holding in the starter.
Since the car will 'bump start' (I presume you mean 'push start'), then you can safely rule out internal engine issues. That only leaves the starter and associated circuits (to include the battery). A bad battery or poor connections to the starter could cause your problem. But if I was a betting man, I'll wager you got a crappy reman starter from the auto parts place.