Even though w69rdy and Emerica answered your question, I don't think you're understanding it, so I'll answer in a different way.
How do I activate/use the special mode?
The answer is simply: You don't activate it, it is a static effect that is applied all the time when you upgrade the Plasma Cutter enough.
It is not a "one-hit kill" effect, it is simply an effect that lights fire to the enemy, causing a bit of extra damage and a small fire animation on the part you shoot.
The plasma cutter is fully upgraded.
Which means that, like in the picture you gave: http://solidlystated.com/software/dead-space-2-plasma-cutter-special-upgrade/, the SPC node at the far right is unlocked, correct? That means that your Plasma Cutter is ready to light up some aliens, no further upgrades or "secret activations" necessary.
The plasma cutter DOES have a special upgrade. Just need to know how to use it.
You are correct, it does have an upgrade, and it is used automatically when you shoot it.
Good Luck and have fun! :)
PS- as for that vid were you saw someone killing people in one hit, you havn't let us see it yet so for now I'll assume the person in the vid was probably playing on easy, or playing some sort of mod. If you could post the vid that would be great.