Printer option two is the easiest to answer :-
A printer that can print customised business cards. By customised I assume you are referring to each card being different by design and / or wording ? Most will do this you just need the correct printer template and pre perforated business card paper.
Option one a printer that can print black text on existing business cards is a potential problem most printers could do this but you may have problems with misdeed's and alignment due to the small paper / card format.
Another solution would be to get a dedicated card printer, quite expensive compared with a standard 'home' printer, and print on the back of you existing cards.