You can stop the draw() continuous callback by issuing -> noLoop();

I don't understand your problem: when you release the mouse button, mouseDragged is no longer called, no?
If your problem is that your disconnected lines are connected, you should write something different, indicating a mouse release, that you must interpret when playing back the record.

Yes, that's the problem- and the question. How to write that break  and have the reader then interpret it. Right now, the reader reads this:

1.  void mouseDragged(){
2.   output.println(mouseX + "$" + mouseY); // Write the coordinate to the file
3.   line(mouseX-lRectX,mouseY-rectRy,pmouseX-lRectX,pmouseY-rectRy);
4. }
5.  
6. void mouseReleased(){
7.  output.println(pmouseX + "#" + pmouseY);
8. }
9.  
10. // Usage
11. if (line.contains("#")) {
12.   String[] pieces = split(line,"#");
13.   int x = int(pieces[0]);
14.   int y = int(pieces[1]);
15.   // Memorize but don't draw the line
16.     px=x;
17.     py=y;
18. } else {
19.   String[] pieces = split(line,"$");
20.   int x = int(pieces[0]);
21.   int y = int(pieces[1]);
22.   line(x,y,px,py);
23.     px=x;
24.     py=y;
25. }

Actually, you can put the px=x lines after the condition, since they are common to both branches. I leave them as is for better understanding...

I suggest to show your full code, or, if too long, to make a simplified version of it, illustrating the issue, so we can run it and see what goes wrong.

I feel like I'm overlooking something obvious here. Help greatly appreciated! The reader is still connecting lines from the last x and y- even when the mouse if not being dragged. That's what I don't want. I think you will see if you run it. Here are the simplified files: 

AWESOME! THANKS! So simple.