UbuAnon:
this looks like bad code writing.
It's certainly not ideal because the blink code will keep checking if the interval has passed, even though there's no point if you only want to set the pin HIGH once. But it is working code that's non-blocking (doesn't use delay()) so it's a very good start. You can proceed to improve it for your specific application.
UbuAnon:
And, pressing the Uno's reset button initiates a new countdown, but I see three rapid blinks, a very short pause, and then one more split-second blink, before the timing interval starts. What is the purpose of these 4 rapid blinks?
Those blinks are done by the bootloader code that runs after every reset. Their purpose is simply to give you a visual indication that the bootloader is running. Of course it's not desirable to have your relay flicking on and off after every reset. You can simply connect the relay to a different pin, as it's only the LED_BUILTIN pin (typically pin 13) that is blinked by the bootloader. You could also not use the bootloader or replace it with a version that doesn't do the blink but those are more advanced operations so I wouldn't recommend them unless you absolutely have no other choice.