To create a point, which is fixed to an object, activate the tool first and then select the object. This new point can be moved via Move Tool, but only within the object.
Note: To put a point in the interior of a Circle or Ellipse you will need to increase the Opacity from 0 first. If you click on the perimeter of an object (e.g. Circle, Ellipse, Polygon), then the point will be fixed to the perimeter rather than the interior.