Before you start adding layers to your puppets, make sure the Default Separate Position is enabled.
With the Default Separate Position preference selected, when you create new peg layers, their X, Y and Z axes are independent from each other.
For cut-out animation, you want to have three independent axes so you can control them individually.
When you lock the three positions together, each time you add a keyframe, it is added on all three positions at the same time.
When you separate the three positions and add a keyframe on one axis, it is not added on the other two. This gives you more freedom when animating with motion keyframes and nudging parts forward and backward over time.