OK, FINALLY got it worked out! (Strange but true answer)
Thanks to yesterday's effort with the VHF's DSC connection, I had discovered / realized that NMEA 0183 is NOT a two-way network, but is instead two parallel one-way networks, sharing a common, well, "Common" wire. One wire is "Out" from a given device; one wire is "In" to that device, and one wire is a common (often erroneously called a "ground") conductor to complete BOTH of those one-way circuits. (Yes, an over-simplification, but one that provides a fairly clear image for analyzing.)
So, the "Out" wire from one device has to go to the "In" wire(s) of any other device(s). Conversely, the "In" wire from that first device has to connect to the "Out" wire of any other device(s). Then, the "Common" wire is, well, common to all of them.
HOWEVER, I also have discovered that some manufacturers, say Garmin, don't use three wires for 0183... they use two ("In" and "Out") and then use the 12VDC negative wire as the common conductor for NMEA 0183. Some, like Standard Horizon, use three wires.
I had been trying to connect the Garmin "Out" to the Raymarine "In" and the Garmin "In" to the Raymarine "Out". I assumed (ASSumed!) that Raymarine must be using the 12VDC negative like Garmin did, so the whole thing should work.
Of course, IT DON"T!!!
So, there had to be some other way to "close the loop".
That got me researching on Raymarine's web site.
Finally found it... "Connect the GPS units NMEA OUT (+) to the Raymarine pilot's NMEA IN (+). Then, to connect negative side, run a line from the Raymarine NMEA IN (-) to the GPS ground wire, or to ground."
The Raymarine just flat doesn't send ANYTHING to the GPS, so it doesn't even use its own "Out"; so, instead (and here is, to me, a VERY weird part), you just put a "Jumper" from the 12VDC negative pin to the NMEA 0183 "Out" pin on the Raymarine connector.
The really bad thing here is that I think I am beginning to really understand NMEA 0183!!!