'Serverless', like many things in our space, is becoming an overloaded term.. but generally what it means is "Functionally, Our architecture does not depend on the provisioning or ongoing maintenance of a server"
The second instance, and is a bit more complicated (and also the one that popularized the term 'serverless'), uses a service like AWS Lambda. Let me explain this by presenting the problem it solves:
Many times in my career I've solved a business problem for a client with little more than some ruby code that performed a periodic extract, transform, and load (typically written as a rake task). Once solved, I'd typically automate it with cron. Then the problem becomes 'where do I host this thing that runs once every hour?' For some clients, we'd set up a server in their existing infrastructure. For others, we'd set up an EC2 instance, even though it was idle 99% of the time. In either of those circumstances, there is a server that requires provisioning, patching, monitoring, updating, etc.
With Amazon Lambda, I can take that rake task and run it on their service as a pure 'function'. I can even schedule it. No longer would that client need a piece of infrastructure for such a simple once-an-hour thing.
With 'serverless' there is still a server, just like with 'cloud' there is still a computer. There is just a level of abstraction on top of it that takes some of the environmental responsibilities for you.