Having your own on premises servers, or hosting your own servers in a data centre is still the "cloud".  Systems are connected to the internet and accessible from anywhere in the world, provided permission and systems are exposed to the internet publicly or via a private connection.

The questions and differences in our opinion revolve around support.

In the on premises server, or your own servers hosted in a data centre solutions, all servers and applications are managed by internal resources or an IT Contractor. In this environment your company provides all the hardware, licencing and so on.

The next level is Infrastructure as a service (IaaS). This is where the core infrastructure, and in most cases; base servers are provided by a hosting provider. The configuration and maintenance of the servers, data and applications are managed by internal resources or an IT Contractor.
 

Platform as a service (PaaS) is where the host provider is responsible for all infrastructure except for data and applications.

And finally Software as a Service (SaaS) where everything is provided and maintained by a host vendor. This is typically your subscription models for Office 365, Azure, website hosts and other online systems.

In each scenario there are pro’s and con’s depending on business requirements, needs and skills.