The situation has clarified a bit, so I'm entering this "new" section.
The world sees my site(s) as it should be. I cannot see what machines outside my home network see. I presume that this is because I have not yet run the ROUTE command.
[However, entering the local IP shows what the public sees via the public IP]
I see two interfaces, corresponding to the two NICs on my system. Both are on the same subnet: 10.1.10.n
Should I continue to use two NICs (I do so to provide separate paths for WAN and LAN)?
I know how to direct all LAN traffic through one NIC, but with both of them on the same subnet, can this work?
My public IP is almost certainly handled completely by my Comcast arrangement. In other words, I never actually see that address, right? It seems to me that if my public IP were 18.104.22.168, then all traffic to that address would end up at my router but that the address would then have no meaning to my home network (which knows only the 10.1.10.n stuff). Is this correct?
Last update, 11 August 2011
Although there are still many questions concerning this event, the bottom line is that Comcast replaced the router (that provides their Internet signal to me) and all works as it did before.
Here is how easy it should be:
1. Comcast supplies the router/modem from which I take my signal.
2. I configure that router to pass all Port 80 traffic to my web server.
Additionally, I must tell "the world" (Network Solutions, in my case) what my IP is. That is provided to me by Comcast.
The situation I started with was much more complex (involving virtual servers, etc), but I had to get it down to the basics in order to get the problem solved.
Thanks for the many suggestions and insights into networking.