More information on my problem.
As an intial matter, the two copies of DG at issue were validated and re-activated for trouble shooting. Didn't fix the problem.
Having no luck with the LAN, we tried DG on the internet (ZehDon's suggestion).
Computer #1 can log into Demigod multiplayer via the internet. Computer #2 cannot. So it's looking like Computer #2 is the "source" of the conflict.
Both computers have Windows 7 - 64 bit, and Trend Micro firewalls/anti-virus. The LAN is a cable connection between the two computers.
Computer #1 has cable connection to internet modem. Computer #2 has wifi to the internet (via the same modem).
Computer #2 has access to the internet, can log into Impulse itself, but when trying to log in DG, it gives us an "invalid user or password." And the message is immediate, as if there's no attempt at connection at all.
This may be a totally irrelevant fact, but I'm going to throw it in case it might have something to do with this. First, the two copies/licenses of DG at issue were bought by the same user (me). When activating DG on each computer, my email had to be used for each (with different product keys). I was reminded of this when I reactivated both copies to troubleshoot this problem. Since there seems to be some goofy login issue, I wanted to mention that.
If anyone has any more ideas, let me know. Thanks for your help.