Wikipedia defines a SaaS (Software-as-a-Service) as software that is licensed on demand. This implies that the end user must pay for the software for the software to be SaaS. I think this definition has changed. Back in the day, they used to call software used by you, but hosted by someone else ASP (for Application Service Provider). That term is no longer used and has been replaced by SaaS.
Often, when people refer to SaaS applications, they point to SalesForce.com - which meets the Wikipedia definition, and Google Docs, which sometimes does. You see, Google docs is free if you only want to use it for personal use and don't care what the URL is. However, they do license it to business on a cost per user basis. Is the free version not SaaS and the paid version SaaS? Technically, Google licenses the application to the free users, so the Wikipedia definition is correct. But Wikipedia also says that is does not matter if SaaS is delivered via ASP or is hosted locally. Again implying that it's the payment model that defines SaaS.
Unfortunately for Wikipedia editors, they will soon have to update this definition. For all practical purposes, the mainstream tech community has changed SaaS to mean what ASP used to mean. Basically, software hosted by a 3rd party and accessed remotely. The payment model is still inherent in the definition, but is secondary to the delivery method.
I once took a class titled "The evolution of the English language." It was a bad class and I took it for a filler credit, but I did learn that the meaning of words change over time and no dictionary can stop it. In this case, I think the meaning has already changed in the common usage, now we just have to update Wikipedia.