THANK YOU AI20! Finally found the steps that worked for those using the MAC version 10.9.5. THIS is how I FINALLY got mine to work. The only correction to the step I had to do to get it to work was the following (see CAPS & BOLD text below).
Steps:
1. Click System Preferences
2. Click iCloud
3. Enter iCloud password (with no code); a notification will get sent to your iPhone or device, click ok.
4. Click left arrow -- (I DID THIS STEP AND IT TOOK ME TO THE SYSTEM PREF. MAIN SCREEN WHERE "ACCOUNT DETAILS" DOESN'T APPEAR. SO I BASICALLY STAYED IN THE iCLOUD SCREEN THAT SHOWS THE MESSAGE TO ENTER VERIFICATION CODE AND I CLICKED "ACCOUNT DETAILS" FROM THERE.)
5. Click Account Details, this will launch https://appleid.apple.com/
6. Enter credentials (iCloud username and password), another notification will get sent to your iPhone or device (this is the one you will use for the verification (password + code) -- (DO NOT USE THE CODE OBTAINED FROM THIS STEP TO ENTER IT ON THE https://appleid.apple.com VERIFICATION PROMPT SCREEN. USE IT WHEN YOU GO BACK TO iCLOUD ON SYSTEM PREF. AND THEN RETYPE YOUR PASSWORD + CODE ALL TOGETHER NO SPACES.)
7. Go back to iCloud and enter your password+code when prompted for verification password
Good luck to all encountering this issue. Love MAC but sometimes simple things get too complicated.