Add the new line as a new path component within the overall shape path.
So, on the example you show above make sure path operations is set to Combine Shapes then with the pen tool Shift click on the left point to create a new path component in the shape - hold the shift key down and click on the right side (holding shift will give a horizontal line)
You can easily do this in Illustrator, since it has a "Snap to Point" feature. You would draw your circle, and de-select it. Then hover the Pen near one of the points, and the point will ":Light up", and the Pen point will snap to it.
I tried this, the result was the creation of another point, i almost couldn't notice due to the proximity between the points.
What i want is to target the settled points with precision.
Ex.: the left and right points coordenates are exactly in (500,750) and (1000,750), when i drag the line near both positions should automatically aim me to the exactly point ( in form of square).
Not sure why you have that extra path point to the left - but you can use the direct selection tool (white arrow) to move it or the pen tool to delete it.
There is nothing in the navigator panel as you just have a path at present - so no pixels have changed. You will need to either stroke the path, fill the path or turn it into a shape (with an associated fill or stroke) for it to show in the navigator panel.
If you have CC2018, try using the Curvature Pen tool
Turn on the grid, and lay down points in Red, Green, blue ord and continue to complete the circle.
Hover over the top point till you see the target icon, and double click.
Repeat for the right hand point.
Note, I am not yet completely up to speed with the Curvature Pen tool, so I might not be using the optimal workflow here, but this does work.