I got it to work! :clap: :clap: :clap:
Short answer: I set ".htaccess Basic Authentication Mode" to False.
Longer explanation: One of the settings in Admin > Modules > Payment > GoogleCheckout is ".htaccess Basic Authentication Mode". It reads:
.htaccess Basic Authentication Mode
Your site Site in installed with PHP over CGI?
This configuration will disable PHP Basic Authentication that is NOT compatible with CGI used in the responsehandler.php to validate Google Checkout messages.
If setted True you MUST configure your .htaccess files here.
Am I the only one finding this confusing?
First, I had to answer the first question: "Your site Site in installed with PHP over CGI?"
I checked with my web host and my site is indeed installed with "PHP over CGI". So, I set the option to True.
Then, I followed the instructions: "If setted True you MUST configure your .htaccess files here."
But this turned out to be a mistake. What I should have done instead was simply set this option (".htaccess Basic Authentication Mode") to False.
I still have no idea how setting this option to False agrees with GCO's directive "Your site Site in installed with PHP over CGI?", given the fact the answer to this question (for my website) is True.
But hey - it works. I can't argue with results. :D