Ok besides the fact that Cpk and Ppk are worse than useless and an abomination on the face of the earth (IMHO), the original definitions (and the one most reputable statistical software uses) are this:

Cpk = the ratio of the process spread to the tolerance using the within subgroup standard deviation. This assumes a homogenous process where the between subgroup variation is nothing more than sampling error. It is also refered to as the 'capability' index or short term capability becuase there is no subgroup to subgroup variation.

Ppk = the ratio of the process spread to the tolerance using the total standard deviation; within subgroup and between subgroup. This assumes a non-homogenous process where the between subgroup variation is more than sampling error. It is also refered to as the 'performance' index or long term capability becuase there is subgroup to subgroup variation.

IF the process is homogenous then Ppk and Cpk will yield very similar numbers...

Non-homogenous processes can be stable and predictable and capable. The appropriate control chart would require a rational subgroup schema different than the traditionally taught subgrouping based on sequential parts. in these cases, yes Ppk can be predictive if the other assumptions are met. (random representative sampling, Normal distribution, yada yada.)

There is a corruption of these original definitions perpetuated by some automotive and aerospace companies where the Ppk formula is used for short term studies in development phases where 30-60 sequential parts are made under the same conditions. (This is a valid method for short term capability.) Some companies will specify the calculation of "Cpk" but provide the formula for Ppk...