If I understand you correctly, you want a button that you press to make it so that you are then drawing with an ellipse?
The way you have it set up, you are only drawing an ellipse when the cursor is inside the button:

1. if ((mouseX > 50)&&(mouseX < 100)&& (mouseY > 50)&&(mouseY < 100)) { //If the mouse is inside the rectangle
   
2.   drawing(); //Call drawing()
   
3.   loop(); //Not really sure why you have this, it's not necessary
   
4. }

You probably want to create a global boolean which holds the value of whether or not you're drawing an ellipse, which, when applied to your code, should look like this:

1. //Declare the boolean
2. boolean drawingEllipse = false;
3. //AND EVEN MORE ADVANCED:
4. //An int set to the value of last mouse press
5. int lastClick = 0;
6. void setup() {
7.   size (500, 500);
8.   smooth();
9.   background (255);
10.   rect(50, 50, 50, 50);
11. }
12. void draw() {
13.   if (mousePressed) {
14.     if ((mouseX > 50)&&(mouseX < 100)&& (mouseY > 50)&&(mouseY < 100) && millis() - lastClick > 400) {
15.       //If the button is clicked, toggle the boolean on / off
16.       drawingEllipse = !drawingEllipse;
17.       lastClick = millis();
18.     }
19.   }
20.   //Just put this in draw(), you don't need a special method
21.   if(drawingEllipse && mousePressed && millis() - lastClick > 400) {
22.     ellipse (mouseX, mouseY, 50, 50);
23.   }
24.   //And finally, an identifier for true / false:
25.   if(drawingEllipse) fill(0);
26.   rect(50, 50, 50, 50);
27.   fill(255);
28. }

And to get even more advanced, with more room for expansion, you could create an int containing the current mode and a set of constants (final int)...

If you want to know how to do this, I could tell you but this might already be overwhelming
I hope this helped