Why does `ps -x | grep foo` include the grep command?

As Bravo points out, a pipe in Linux is not a file, it is dynamic. So ps -x | grep login will actually start both programs at once, so that one can pitch down the pipe and the other can catch. The reason it is not deterministic, why your contrived example doesn't always show four grep instances, is that an instance may not be started until there is something in the pipe for it to do; or the instance in question may not be fully started when ps reads the process list.

Unix is a multitasking operating system. It doesn't start ps first or last. It starts all the things in the pipeline in parallel. Some might finish first or start after ps starts, and that's where your race comes from that makes the results slightly non-deterministic.

Basically, there is no order of operations here. A pipeline is a data stream, not a math formula.

The beauty of the pipe in unix is that the second program can start processing the input before the first program finishes, and thus it isn't necessary to collect and store the entire data stream in memory (or disk) before giving it to the next thing in the pipe.