I was taught that there are two phases of the engagement. First, where you use your fingers and move the point to search out your opponent's blade, making contact with it. This part is quick and fluid, and your shoulder must be relaxed. As you make contact with their blade, the second phase is where you complete the extension of your arm and push forwards. This part is where you also block out your opponent's point - in French, they say "fermes la porte" ("close the door") - for example, if you engage in sixte (esp. against a lefty) then it's best to close the door by lifting your hand.
As for quarte, in my opinion the best parry/engagement is done with supination (i.e. palm facing upwards, just slightly). This gives a couple of important advantages when compared to rolling your wrist over:
(i) your point stays in line with your opponent; if you turn your wrist, the first thing that happens is displacement of your point.
(ii) greater flexibility in the wrist/elbow, allowing for an easier (and therefore quicker) riposte; roll over your wrist, and it is much more difficult to riposte.
(iii) parrying with your palm facing upwards is a more efficient movement, and is therefore quicker.
The first two points are particularly important in épée, where detached movements are suicide.
I hope this helps
