The only thing about your question that stands out as wrong is that you say
A would run first, then B gets the stdout of A
In fact, both programs would be started at pretty much the same time. If there's no input for
B when it tries to read, it will block until there is input to read. Likewise, if there's nobody reading the output from
A, its writes will block until its output is read (some of it will be buffered by the pipe).
The only thing synchronising the processes that take part in a pipeline is the I/O, i.e. the reading and writing. If no writing or reading happens, then the two processes will run totally independent of each other. If one ignores the reading or writing of the other, the ignored process will block and eventually be killed by a
SIGPIPE signal (if writing) or get an end-of-file condition on its standard input stream (if reading) when the other process terminates.
The idiomatic way to describe
A | B is that it's a pipeline containing two programs. The output produced on standard output from the first program is available to be read on the standard input by the second ("[the output of]
A is piped into [the input of]
B"). The shell does the required plumbing to allow this to happen.
If you want to use the words "consumer" and "producer", I suppose that's ok too.
The fact that these are programs written in C is not relevant. The fact that this is Linux, macOS, OpenBSD or AIX is not relevant.