When you create an account on Visual Studio Team Services, you have accessed to the new Inheritance process model. This process model has a nice "what you see is what you get" (WYSIWYG) editing experience and includes the concept of "sharing." With sharing, when you modify the process, all the team projects that are based on the process automatically get these changes.
Microsoft Team Foundation Server (TFS, on-premise version) uses the "on-prem XML" process model. As the name suggests, this process model is based on a set of XML files. However, this process model doesn't have the concept of sharing. When the user creates a team project, the process metadata is copied into the team project. To change the process, the user must change the metadata in the team project. Modifications to the process template itself are applied to new projects only. That's also why it’s "Process Templates" in TFS and "Processes" in VSTS.
The third process model is available only for those who have used the TFS Database Import Service. This process model is called the Hosted XML process model. This process model uses XML files to define the process, such as the "on-prem XML" process model. As opposed to the TFS version, Hosted XML supports the concept of "sharing," such as the Inheritance process model. The difference between Inheritance and Hosted XML is where the process metadata is stored. In the Inheritance process model, team projects read their process metadata directly from the process. However, team projects still store the process metadata locally in Hosted XML. To support the concept of sharing in Hosted XML, the process metadata is copied to each team project when an Hosted XML process is changed.