I've worked on a number of systems that people referred to as 'beowulf clusters,' but not a single one was actually running the Beowulf software.
I see... so that's not a car, it's an automobile!!
I think you pedanted yourself right out of making any sense. What you describe are indeed Beowulf clusters.
No particular piece of software defines a cluster as a Beowulf. Beowulf clusters normally run a Unix-like operating system, such as BSD, Linux, or Solaris, normally built from free and open source software. Commonly used parallel processing libraries include Message Passing Interface (MPI) and Parallel Virtual Machine (PVM). Both of these permit the programmer to divide a task among a group of networked computers, and collect the results of processing. Examples of MPI software include OpenMPI or MPICH. There are additional MPI implementations available.