What does the subscript 0 mean on terms like $\mathsf{ATR}_0$? Does it mean the same thing in $\Pi^1_k\text{}\mathsf{CA}_0$?
If I frame higher order analogues of these, should I change that subscript?
What does the subscript 0 mean on terms like $\mathsf{ATR}_0$? Does it mean the same thing in $\Pi^1_k\text{}\mathsf{CA}_0$? If I frame higher order analogues of these, should I change that subscript? 


As the other answer points out, the subscript 0 means restricted induction. However, without the subscript 0, there are two conventions:
For higherorder analogues, it is still a question whether restricted induction or full induction is included. Kohlenbach [1] has used notation such as $\mathsf{ACA}_0^\omega$ to refer to the analogue of $\mathsf{ACA}_0 $ formalized in arithmetic in all finite types. In this context, though, there are many different ways in which induction can be restricted. So notation like $\widehat{\mathsf{E\text{}HA}}^\omega_\upharpoonright $ is used in the literature, where the hat and the harpoon refer to different sorts of restrictions. These notations are explained in Kohlenbach's Applied Proof Theory or in Troelstra's Metamathematical Investigations. 1: Ulrich Kohlenbach, "Higher Order Reverse Mathematics", Reverse Mathematics 2001, Lecture Notes in Logic, 2005, ftp://ftp.daimi.au.dk/BRICS/RS/00/49/BRICSRS0049.pdf 


The subscript $0$ is meant to indicate the amount of induction that the theory has. The wikipedia entry on Reverse mathematics says of the big five theories of reverse mathematics that


