First, you should get a warning when you are animating paths that deleting or adding a point will cause unpredictable results. It's not generally a good idea to delete path points.
Checking Preserve Constant Vertex and Feather Count in the General Preferences will help avoid these problems. With this option checked when you delete all but one point it will foul up your animation before that point because now there are no points to preserve. You will end up with one point which does not define a line or a shape.
If you decide to uncheck Preserve Constant Vertex and Feather Count you can do what you want but you'll have to open the path before you delete the next to the last point. Once a closed path always a closed. This is why you are getting the loop.
You must have two points to open or close a path. The work around is to open the path before you delete the next to the last point. Do that by selecting the last two points in your shape (or any two points) and right clicking and then select Mask and Shape Path > Closed and uncheck Closed. Then delete the point. Now when you start adding points again you will have an open path.
Be aware however, if you have Fill selected then you will get a fill that extends between the points on your path.
There's no closed path here but the fill will still render. There's no way around that. I hope this helps you understand editing and animating paths.
The other option would be to just start another shape layer.
Thank you so much Rick! I didn't know that you had to select two points to open it.
I have a couple of questions.
1. If I don't delete the path points, does that mean I have to move every single one individually? How would that work if I was trying to make a big object into a smaller object?
1. You said that another option would get to start another shape layer. How would that work?
You can select all of the points and then use transform. You can also hold down the Command/Ctrl key with the path tool selected and marquis select multiple text points to animate transform them. Take a look at this short roto sample. You can see how I'm using the marquis tool to select multiple points. When multiple points are selected you can then press Ctrl/Cmnd + T to open up the transform tool and scale or rotate them.