> - The demands of a consumer Web site/API or multitenant enterprise application simply exceed the computing capacity of any one machine.
> - An enterprise moves an existing application, such as a three-tier system, onto a cloud service provider in order to save on hardware/data-center costs.
When you've exhausted the capacity of a single machine, typically you don't jump straight to a distributed system. You can scale out horizontally with a stateless application layer, as long as the data storage on the backend can handle all the load. You can also scale database reads horizontally, using read replicas.
This horizontal scale-out is not a distributed system, since consensus ("source of truth") still lives in one machine.
So I think a better phrasing of #1 would be "When your write patterns exceed the computing capacity of any one machine".