However, whenever I tried to click on the stopwatch to create a keyframe, I was unable to move a particular vertex away from its original position, without moving the whole mask shape.
the stopwatch is only required for the first keyframe, if you click on it once again you will remove all keyframes.
I don't see in the screenshots that you have any keyframes for the mask path.
set a keyframe at the first position where the path is in its start position by activating the stopwatch - you are required to do this only for the first keyframe, now advance your CTI later in time and move your path to its ending position - this will set another keyframe.
BTW, if you want the reveal to move in an angle you can use the linear wipe effect instead of animating a mask path, it might be more elegant.
Here's a very short video on how to animate a path in AE:
It should help.
Any time you click on the stopwatch or highlight a keyframe of a mask path (or shape path), ALL vertices will be selected in your viewer window. You then have to click and drag a selection box around whatever vertex you want to edit so that the rest get de-selected. Some people find this behavior confusing, but I find it to be very convenient, as long as you are aware that this functionality exists. If you want to activate your mask or shape path WITHOUT selecting all the vertices, click on the line just above the Path line (the one just above where the stopwatch is) and your comp will shift focus to the path vertices, allowing you to simply click (rather than click and drag) a single vertex.