But surely a process model is whatever it is supposed to represent? By that I mean 'Can't we put whatever needs to be put into a process model to make it do the job it's supposed to?'
The short answer is "Yes"
The long answer is "Absolutely not!"
If you're putting together some sort of diagram that can explain or illustrate something to users (whether these are existing users or potential new users), you need to make the diagram as clear and informative as possible. This means adding in (or subtracting) whatever information will help you achieve that. In many cases this means a process modeling package is not the appropriate tool. Visio probably is.
I've spoken before on this site about the use of Visio for process modeling (and, incidentally there is a section on this in the new eBook I am writing 'The Perfect Process Project"). To recap, my advise is "Don't use Visio for Process Modeling".
However, as a communication device that can show lots of unconnected things in a clear, visual manner, Visio is an excellent tool. Remember, though, that it is a drawing tool not a modeling tool. The diagram you draw in Visio should be based on an existing carefully modeled process from within the modeling tool of your choice.
With Visio you can add in little figures to represent your users, you can format the model to ignore good process notation standards, you can drop big chunks of descriptive text all over your diagram. In short it is the correct tool for putting something out there that is informative without being totally accurate (and by accurate I mean referring to established notation standards).
However I would advise against this.
The reason is simple: Expectation setting.
If you make it clear or known to your users that within your project it is acceptable to produce non-standard process models, then this expectation will then have to be met and managed. Similarly, if you make it known to your project that it is acceptable to define and manage process models in a drawing tool rather than a modeling tool then this expectation will also have to be met and managed.
It's all very easy to say "I know Visio, I can use it. Everyone in my organisation can use it and it's cheap (at least compared with most Process Modeling tools on the market at the moment)" But remember the objective of your project is to define & manage your processes as well as to produce the pictures that illustrate these.
Experience has taught me that letting inexperienced users draw their 'processes' in Visio is a recipe for disaster.
Don't do it.
You will end up with non-standard diagrams, superfluous objects, lack of consistency, missing notation standards and other similar issues.
You will also end up with a set of pieces of paper that are not connected, diagrams that don't link, activities that are repeated but named differently, deliverables that are missing from one diagram but appear in other diagrams and a myriad of similar issues.
Don't do it!