I believe this has something to do with MAC address of the router, as the modem stores such information and makes a lease query based off that address (the modem itself most likely doesn't receive own external IP, yet in most cases accessible via web-interface - i.e. http://192.168.100.1
for Motorola models, google for your make. For this, I presume, you must be directly connected to the modem with one of your systems). One of the ways to resolve it would be spoofing router's MAC (if possible at all), another - reseting the cable modem to the factory default to clear MACs stored in modem's memory, which is easier to do via web-interface, but I wouldn't recommend it without notifying and hence receiving a blessing from your ISP first, as I'm not a professional and have only a vague understanding of the consequences, moreover never tried it before myself. As for what can you do to make the IP release/renewal easier next time - as far as I understand there is nothing to be done on the user side except what was already mentioned: reseting to factory defaults or spoofing router's MAC, which is probably not that easy to do, if that option is not openly available on your model.