I believe the issue might be related to the physical connection, it would be of great help if you could add a schematic and pictures of how you are connecting the three parts. I doubt this is related to the sketch part but if you had access to a logic analyzer, it would be great to check the outputs of all the stages of the "translation".
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Yes, I found it is was related to the physical connection!
In Scenario 3, I originally had the Sparkfun TTL/RS232 adapter that had a female DB9 connector wired to the HDMI switch female DB9 connector using a male/male serial cable. The serial cable I bought was a straight through cable.
I swapped it out with a null modem cable and it worked right away.
Looking at the wiring diagram, I thought swapping the Tx/Rx pins on the Sparkfun adapter would have achieved the same result as the null modem cable, but apparently not.
If anybody can point out why the null modem cable made the difference, that would be something good to know for the future.