Slavery as a punishment for a crime is still slavery.

Indeed, but it depends how broad your definition of slavery is.

You're comparing this kind of slavery with chattel slavery to draw such conclusions, but in reality there are multiple systems that can be called slavery.

I'm aware there were other forms of slavery, but chattel slavery is the type generally referred to. I guess under other types of slavery penal labor would in fact classify as slavery and/or indentured servitude.

The issue I have with leasing/selling slaves to private companies is that two problems then arise: those slaves could be taking jobs that normal citizens want or need, and only creating a few (those few jobs created being those looking after or otherwise caring for said slaves) and the possibility of escape.

Ideally penal labor would be used to perform unpleasant or unprofitable jobs that normal workers would never reasonably want or be expected to do. Take for instance some types of farming that, without significant automation, are very unprofitable; unpaid labor eliminates this issue and allows these products to be produced and go to market without concerns for losing money from having to pay workers.