Is there a difference in the meaning of power and current amplifier terms?
If you ask 10 EEs this, you will probably get close to 10 answers.  This is not a well-defined convention, but is roughly outlined as you guessed.  A power amplifier will be able to provide more power than the previous stage.  A current amplifier will be able to provide more current than the previous stage.

If you need a definitive answer, I would say ask the person that you are involved with on this project.  The key is always to understand what the other person means, not necessarily what they say.  Basic communication theory.  :-)