Amazon itself made its definitions of this architecture here (my emphasis)
build and run applications without [having to] think about on servers
It is not that it is without servers, it is that the focus on it will be less.
Amazon Lambda is in vogue when we talk about serverless and AWS because you write code without worrying about which server, container or virtual machine it will run.
The big gain is in the development of the application, since the focus will be on it and not on server configuration and these Devops blah blah blah blah. Worth a read on noops culture. From the AWS website, same page I quoted earlier:
decrease [server management] overhead enables developers to have the energy and time to spend developing good products that scale and are reliable
There is no way to apply everything using this architecture, there are cases (maybe most), that the best would be to manage the server, the scalability and the availability of the application.
In the definition you brought from Wikipedia, I can emphasize
is a cloud computing Execution model in which the cloud Provider dynamically manages the allocation of machine Resources
It’s Azure, AWS or Google Cloud Platform that would manage the server on issues of scalability, resource allocation, location and so on, while you and your team would focus on application development.
It’s an interesting development paradigm, but obviously there are controversies. It’s good to be in control too. Question everything!
"A human being undoubtedly does not evolve. Questioning is the first step in opening ourselves to the new." - Roberto Shinyashiki
I guess we’re forced to do stateless like that, huh? When you manage the application server, there are people who are doing something that turns out to be stateful.
– Jefferson Quesado