According to the NIST definition of cloud computing1, there are three service models for cloud computing: Infrastructure as a Service (IaaS), Platform as a Service (PaaS), and Software as a Service (SaaS). In the SaaS model, the cloud provider delivers the software applications over the internet, and the users access them from various devices through a web browser or a program interface. The cloud provider manages the underlying infrastructure, including the servers, databases, and code of the applications. The users do not need to install, update, or maintain the software, and they only pay for the service they use. The scenario described in the question is an example of the SaaS model, as the user is provided access over the internet to an application running on a cloud infrastructure, and the vendor hosts and maintains the servers, databases, and code of that application. Reference:
SP 800-145, The NIST Definition of Cloud Computing | CSRC
Final Version of NIST Cloud Computing Definition Published
NIST Cloud Computing Program - NCCP | NIST
SaaS - User responsible for only the data, vendor responsible for rest