Finally got something useful done with the Thinkpad with the broken backlight. Thanks to lots of help from Paul, and a critical bit of advice from Stephen, it’s now living on my network and visible to the outside world.
What had me initially confused was that both my modem (a SpeedTouch 546) and my Netgear router have NAT firewalls. I had to declare the router as a DMZ on my modem, and the Thinkpad a DMZ on my router. Also, the router’s DynDNS support was only reporting its IP address as seen behind the modem, so I had to turn that off and use dynDNS from the modem.
Security hole? Perhaps; but it’s not as if OpenBSD is the least secure or most widely-used OS. I’ve really only got sshd and thttpd running, so there’s not much to chew on …