Quick twitch exercises are typically done with lighter weights
Exercises like squats, dead lifts, lunges, calf raises and hamstring curls using free weights (not machines) work out both the primary muscle and the stabilizers and in the process they work out the quick twitch muscles making them stronger and making the person more explosive. Strengthening your quick twitch muscles with heavy lifting improves power output, so when you use your quick twitch muscles, they push you harder.
There is more than one way to skin that cat.
Well, yes really.
In order to improve your power output, the exercise needs to be done quickly (hence, the velocity aspect which I included previously). Heavy lifting does not automatically improve power output. If it is done with a slow velocity, the power output suffers (I can give examples if you like). I'm sorry, but that is a fact. There is no sweet talking around that.
When I said "light" I did not mean to imply incredibly light if thats what you interpreted. Typically, exercises that aim to improve power include weights that are 30-40% of a 1RM (I'm sure you can find slight variance with that number.
So if you wanted to gain power through those exercises you mentioned like squats, dead lifts, lunges, you would go "lower" in weight to allow you to be explosive. This is still challenging the muscles obviously. If it is done slowly, you are more so working on improving strength, not power.
Olympic lifts, such as a clean, snatch, etc are an "explosive" lift and are some examples of power lifts. They have to be done with a weight which you can "explode" through the motion. If it is too heavy, they can't be done properly
The whole point of my previous post was that if you train slow and just lift heavy weights to add bulk to your frame, you don't automatically stay or become quick. You may become strong, which is good, but there is not a direct correlation between that and the term power.
And also, there actually are machines (look up Keiser) that can measure power output. It is actually the main purpose of them and they are probably the best at measuring it. I did not bring up the use of machines, but since you mentioned it, wanted to put that out there.