Need difference between power rails,voltage rail and current rail
They are probably all the same but different people in different circumstances call them different names to suit what they are trying to achieve or describe.

For instance, I would use "power rail" or "voltage rail" without thinking it would confuse anyone. The basic "voltage rail" implies the connections to a voltage source but, when you start drawing current (via those rails) from the voltage source, you are (likely) also drawing power. P = V*I.

So suddenly, a voltage rail has also become a power rail.

"Current rail" is a bit outside my vocabulary because I would only use the terms voltage or power rails but, if a voltage rail is passing current, it could be described as a current rail but, I would prefer to call it a power or voltage rail.