When you select Add Point on Line, you will be prompted to select an existing line. 

You have two options:

On the Line - Choose this to put your new point somewhere from the start point and end point of the selected line.

Beyond the Line - Choose this to put your new point on the same vector as the line, but before the start point or after the end point.

On the Line

By default, the point will placed in the center of the selected line.

You can move the point from the center position in a number of ways. Regardless of which control you use, all controls will update.

Slider

There is a slider at the top that will move the point up and down the line. Move the slider and release it to see the exact placement.

From Start / End

You can designate the precise placement from the start point or the end point of the line. Fill in a value in either box.

Offset

You can choose to offset your point from the line, too. Enter a value to offset and you'll see the point jump off the line perpendicular to it. If you want it on the other side of the line, select Offset to Other Side.

Select Save to keep the new point.

Beyond the Line

You can choose whether the point is created beyond the start point or the end point of the line. Enter a value in the Beyond box and then choose the Start or End button.