This thing becomes even weirder. So I got myself my own cable modem on their "approved list", called up Time Warner who promptly "activated" it and took off the modem rental charge. However, it turns out NONE of the non-Time Warner cable modems support digital phone hookup that Time Warner uses. So even if you get your own modem, if you have TW digital phone service, you STILL have to have TW cable modem!
They suggested I go to the local TW center to pick up a "free cable splitter" so that I can have BOTH their modem and my modem hooked up at the same time, with TW modem only hooked up to the phone line. I didn't have the time to go pick up the spliter, but lo and behold, even without my modem hooked up, the whole cable/phone works fine via TW modem like before, without my modem even hooked up.
This is incredible nonsense and a pathetic attempt by TW to raise prices. On the bright side, I now have my own modem sitting around without wasting power (?) and the rental charge erased from my account