Don't just use your thumbs... Use your whole palm to twist it, with maybe a thumb in one of the recesses, applying slight downward pressure.
I successfully completed this for a Mac Mini Late 2012 purchased at the end of Feb 2014, replacing the stock 500gb snail drive with a 500gb Samsung 840 Series SSD. No logic board removal was required, and the antenna could remain attached - there was actually a plastic cover over the antenna connector, meaning removal and replacement would've been a pain anyway.
This replacement was a breeze, and even though I took my time and was every careful, it was only a 20 minute job. The only thing that caused me pause was the removal of the tape holding the old drive's protective sheet onto the SATA connector... Trying to do that without tearing and creasing was not fun, though my iFixit spudger helped.
I just want to add that I think it's criminal that this very capable little Mac is still sold with the antiquated likes of a 5400rpm drive. My intention on the outset of purchase was to upgrade to an SSD, but I was shocked at how slow the stock drive made the Mac run, with countless slowdowns and lockups.
The cable is a standard length, but I feel it was too long as it needed to be folded to fit. A 10 inch cable will be fine.
I can confirm that you do not need to remove the power supply to do this - you can safely skip these steps! Anyway, this install went well for me, not as scary as I thought it would be. You just need to be prepared, remain calm and keep track of all the screws!
Can anybody explain why the PSU module needs to be removed (steps 34-37)? I saw a similar guide elsewhere and it skipped this completely. Is this not required? If so, that might explain why this guide says that to re-assemble do the reverse from step 33 and not 38 as expected!