Every time I come across something like "process 0 does x task" , I am inclined to think they mean processor.
After reading a bit more about it, I find that there are two memory classifications, shared memory and distributed memory: A shared memory executes something like a thread (implying same data is available to all processors- hence it makes sense to call it a process) However, even for distributed memory it is called a process instead of a processor. For example: "Process 0 is computing the partial dot product"
Why is this so? Why is it called a process and not a processor?
PS. I hope this question is not trivial :)
These other answers are all pretty spot on. Processors are physical, processes are software. So a quad core CPU will have 4 processors, but can run many more processes.
Your confusion around distributed terminology is fair though. In distributed computing, typically X number of processes will be executed equal to the number of hardware processors. In this scenario, each processes gets an ID in software often called a rank. Ranks are independent of processors, and different ranks will have different tasks. So when you report a status, information is relative to the process rank, and not the physical processor.
To rephrase, in distributed computing there will usually be one process running on each processor. The process will have a unique id that is more important in the software than the physical processor it is running on, so status information is given about the process. As the number of processes and processors are equal, this distinction can get a bit blurred.