When I think of Games as a Service, I think of things like:

"The crux of the Newell's address focused on the concept that direct communications with customers, transparency, and constant updates are the best ways to maximize profits from a product. In this way, Valve views its products as a service rather than a finished project. When the company shipped Team Fortress 2, work wasn't done. Rather, the team said, "Now we can start." The team has then gone on to ship 63 updates -- which include anything from bug fixes to new game modes -- to the game in just over 14 months. This can directly result in increased sales that would normally taper off over time. As Newell put it, "When you want to promote your product, you're going to use your customers to reach new customers." " https://www.ign.com/articles/2009/02/19/dice-2009-keynote-gabe-newell

Games as a Service I think of as an overarching concept based around the idea of service not stopping at the point of sale. After that, the different approaches are almost "sub-classifications".

There are those games that are touted as "Live Services" - when I use that term, I think of games that provide ongoing content, and maintains the game servers in exchange for varying streams of income. These are games that will typically "stop working" when the official servers go down. I consider games like Anthem and Loadout to be examples of Live Services in this respect. Games like World of Warcraft I consider Live Services, but I go one further and call those "subscription services", since they require the subscription to play and in theory this is the main income that funds the game staying online.

The way I see it, all Live/Subscription Services fall under the Games as a Service banner, but not all Games as a Service will necessarily be either. To me, it's more nuanced than one general classification, especially with the way things blend together these days; games can have multiple income streams (subscription, microtransactions, battlepasses, season passes, so on), as well as multiple forms of content delivery (free updates, expansions, DLC).

The lines have been blurred further with "early access" and "incomplete" titles being released that have constant content updates simply to get to a release state. However, these types of games have those stars that typically shine through. Such as No Mans Sky or (contentiously) Fallout 76.

tl;dr To me GaaS is the literal idea of treating games as more than a one-time product, but evolution in how content is delivered and monetized have lead to many different approaches. Unfortunately, the normalized ones (the maliciously monetized and despair-inducing) are so far apart from the "good ones" like L4D2, that it is difficult to consider that they are both actually examples of "Games as a Service".

It's long and I'm sorry.