Max Wideman Intervenes with Other Suggestions
@Larry: Your remarks are right on. Indeed, the HRIS/Payroll example could be conceived as three separate projects, but I rather think that this could be unwise, because they are really different management levels of the same structure, but then, no longer parts of the one original project.
However, I am sure someone will come in with saying that a project could be defined as encompassing all three levels. An example: design, build and run a "project/product" through its life span right up to its disposal. This example could be in mining where the raw material is known to have a limited life, and the project starts with securing the required investment, building a management team, etc., etc. through to final environmental restoration. But for purposes of teaching and training it is much better to keep it simple, and to basic elements on which more sophisticated management assemblies can be built.
By the way, while I have used a "physical" example here, software can be envisioned in the same way.
@Larry: In the project example you outlined (i.e. "Let's say that Company A needs a new, integrated Human Resources and Payroll Management system"), senior management may insist on the "One Project" title. They may do this for corporate accountability or financial reasons. Then a better way is to establish an appropriate number of sequential "Major Phases", each with a Go/No-Go gate before moving on.
That is in effect a series of sub-projects that achieves the same purpose as you have outlined. Of course an even better approach is to establish the whole exercise as a "Program". But if the organization is not familiar with Program Management (as many aren't) then that approach is a non-starter.