What it all came down to was that merely connecting a DLINK DHP303 Powerline adapter to a LAN port on the Dir-820L killed the 820's Wi-Fi. It switched from Connected to LImited, which made that network useless. I didn't even need to complete the connection by turning on the other DHP-303. Just plugging in the first caused the problem.
So, I am now assuming the DHP-303 is not compatible with the 820L, even though Dlink says that the 820L is backwards compatible with all other devices. Of course, it might still be that there is some parameter in the 820L that I set up wrong but I doubt that. After all, everything works if I connect the routers using an Ethernet cable rather than a DHP-303.
The final workaround was easy. I connected the DHP303 to the LAN port of the slave Dir-655 (which itself is connected to the 820 by an Ethernet cable). Everything worked fine. The 655's network did not go into LImited mode. The 820's network stayed connected too. Indeed, all 3 networks from all 3 routers hummed along happily.
The DHP-303 was obsoleted by DLINK in 2012. Perhaps new Powerlink models might work with the 820. But I now have no need to pursue that expensive option.