A couple of important details are not entirely clear to me.
... After erasing the drive (which is not SSD), tried reinstalling Mavericks but could not proceed, therefore, I used Time Machine to reboot.
If you booted from the TM backup drive, then it invoked its version of Recovery — not the one resident on the (presumably erased) Mini's startup volume. Time Machine backup volumes are bootable only in Recovery.
The reason for asking is that it's not entirely clear to me whether you restored that Time Machine backup or not. Obviously that's not what you would do if you were selling, donating, or otherwise parting with that Mac. You boot the TM backup drive and use its version of Disk Utility to erase the Mac. I assumed that is what you did but I don't want to assume. As an aside, restoring a Time Machine backup also restores the local startup drive's version of Recovery. This is only applicable to those older Macs.
Next important question: Did you encrypt that Mini's startup drive with FileVault? If so you're done. Its data are utterly irretrievable without that FileVault key — which gets erased along with everything else. It's useless.
On the other hand if you did not use FileVault its data are theoretically retrievable, depending on what method you used to erase it and the amount of time effort and expense someone wanted to implement toward that effort.
Lastly, why did you not simply boot from that Mac's original System Install DVD? Those grey discs must accompany the Macs they shipped with from cradle to grave. To properly transfer ownership to a new owner, it's required to reinstall that OS X version. Its new owner would be greeted with the "Hello" language selection screen the same as you did upon initially setting it up.
Personally I would not be interested in buying any Mac that required those discs, unless the seller included them, or unless I happened to have my own copy for that specific model.
That Mac is too old to use Internet Recovery.