12-volt battery with 6-volt generator ??

My Ford 640 has been running down the battery - for the month or so that I've owned it. I keep charging the battery and everything goes fine for a few days. This battery is a definitely a 12-volt, negative ground battery (still got the WalMart stickers on it. Also, my voltmeter shows it to be 12-v, neg-grnd). The other day I realized (yes...I'm a little SLOW on this) that it's running a 6-volt generator. I've attached a picture to get some expert opinions - that's definitely a 6-volt, positive-ground generator, isn't it? I've checked on the Vintage Tractor page and it indicates that this model of tractor should be 6-volt, positive-ground. If so, is this a case of someone (previous owner) who got part-way through a conversion to a 12-volt system, then quit. I think the generator's not working - because I can disconnect the battery while the tractor's running and it will die. I guess I'm trying to find out if anyone's seen anything like this before, or if anyone has any ideas for what I need to do to fix the electrical system. (This might sound stupid, but can I just replace the 12-volt battery with a 6-volt? Isn't there a problem with the negative-ground vs. positive-ground systems?)

Bill Stanley