Okay my memory about the relay names was *wrong.*.
As you can see from the pictures below, there IS in fact the LIGHTS relay, but the other two are MAIN and START.
So when I swapped the two relays, it was the MAIN and not a Turn Signal relay used in the swap.
So I switched the MAIN and the LIGHTS relays.
The upper-left relay socket -- where the LIGHTS relay plugs in -- that is the problem.
When I put the MAIN relay into the LIGHTS socket, all of a sudden the MAIN relay started with this audible buzzing, and vibrating.
And putting the LIGHTS relay into the MAIN relay's socket did not cause any issues. AND it did not buzz or vibrate when it was plugged into the MAIN relay's socket.
Need that freaking wiring diagram.
FUSE BOX AND THE 3 SILVER RELAYS
LAYOUT OF THE FUSE BOX AND RELAYS (this is the black plastic cover of the fuse box with a label on it)
I REMOVED THE 'Lights' AND 'Main' RELAYS -- THEIR SOCKETS ARE THE SAME, JUST 180 DEGREES FROM EACH OTHER
'Lights' AND 'Main' RELAY ARE THE SAME PART
SO IT'S OKAY TO SWAP THEM, WHICH I DID -- NO CHANGE, THUS THE 'Lights' RELAY IS NOT THE PROBLEM
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