I managed to delete a long reply I was writing, and my wife has already gone up to bed, so this is going to be brief.
Yes it is difficult becuase you are going way outside the bounds of normal home use. The home WiFi software developers and home router manufactures do not build in simple setups for complex situations such as yours. Sorry, but that is just the way it is. Networking is hard when you start "coloring outside the lines".
At this point, I do not know what is working.
Can the neighbor talking Mac access the Internet? If not you have to get that working first.
If the neighbor talking Mac can access the internet, then enable internet sharing via Airport, and see if the 2nd Mac can connect via Airport.
Does that work? If not, what is the 2nd Mac's network setup, including IP address, net mask, router address, and DNS server addresses?
I am keeping the Linksys router out of this, as it is apparently just confusing things. Besides the less hardware involved the easier it is to debug.
If sharing via Airport works, you might just declare victory.
If you really want to add the Linksys to the mix, then the next step is to verify ethernet BUT WITHOUT the Linksys.
Switch the neighbor talking Mac so it is sharing via Ethernet.
Connect the 2nd Mac to the neighbor talking Mac via an Ethernet cable.
Does that work? If not, what is the 2nd Mac's network setup, including IP address, net mask, router address, and DNS server addresses?
If that does not work, try a different Ethernet cable. It would not be the first time a bad Ethernet cable has caused networking problems.
If neighbor talking Mac to 2nd Mac via Ethernet works, then and ONLY then would I think about adding the Linksys to the mix.
First, I would use the Linksys in its default mode. That is connect the neighbor talking Mac's Ethernet cable to the blue "Internet" port (or so the WRT54GX-v2 PDF manual says it is). I would have Linksys DHCP enabled (this too would be default Linksys mode). As far as the Linksys is concerned, the neighbor talking Mac is the Linksys' ISP. To the Linksys it does not matter what the neighbor's ISP is, just the signal coming from the neighbor talking Mac.
Does this work. If not, what is the Linksys IP address, net mask, router address, and DNS servers?
It is important to make sure each leg of the network connection is working, which is why I have tried to set things up step by step with tests to make sure each leg works before moving on to the next.
OK, my Wife just told me to go to bed. Good night.