You can't move the ethernet cable from the modem to the other router without a reset. This is well known and I decline to document it here.
But I found most of the time I can power down the cable modem, move the ethernet cable and power up the cable modem to get it to work.
I recently purchased a new Cisco E2000 router and updated it with the latest firmware. When I set this up at home I was able to connect to all computers on my LAN, but did not have an internet connection. I tried renewing the IP on the router, but could not get an internet IP. I called my ISP and they reset my modem which allowed my new router to obtain an IP and I had internet for a short period of time. When my DHCP lease is up the router does not renew the IP and I lose the interwebs. I've tried manually renewing the IP and can't obtain a new IP unless I contact my ISP and have them reset the modem. If I connect my old wired router, however, it immediately gets an IP and renews the IP properly. Any idea why I can't get my new router to properly obtain an internet IP?