Go to the tools panel of Photoshop and select either the add anchor point tool or the delete, basically they are the same tools even though they have different names. You can find the tool beneath the pen.
With the tool, hover over a point, any point in that path and you will see that it changes to the direct selection tool and you can move that anchor point to a different location.
If you hover the line then the cursor will change and show a + sign and you can then click the line to add a new point to that line. As soon as you have added it, again you will then see the direction selection tool and you can then shift that point. You can repeat this over and over along the line and add as many as you wish.
With the delete tool, you can hover over the line and you can do nothing. Hover over a point and the cursor will change to display a - sign and you can then click that point to remove it. You will notice that the path will probably change radically, though this depends on the number of points, curve etc.
If you hold down the option / alt key on your keyboard, the cursor will change to a + sign and then you add a new point with this as long as you are hovering over the line and not a point (though weirdly the cursor suggests that you can click it, it has no effect).
There are no other options for the tools. However, you can constrain the direct selection feature to 45 degrees and 90 degrees etc by holding down the shift key
It should be noted that a combination of the add / delete means that you can change the path in countless ways and create a new path very quickly and this can be selected and then via the edit menu and define custom shape command be saved as a new custom shape and the preset used later.
You can also use the feature with open (such as a line) as well as a closed path (such as a circle). With the open line, you cannot add to the ends of the path but you can remove from the ends using the delete tool in Photoshop