I have always found it very difficult to understand Microsoft Project provisioning options. This video makes it very clear.
At my current client we have Office 365. I have purchased a single copy of Microsoft Project Professional and because of this our Office 365 tenant gets a Project Online licence which allows us to use the Project Web App (in history and on-premises this is called “Project Server”).
Although I have bought only one Microsoft Project Professional licence we can provision as many Project Web App sites in SharePoint as we want. This video also shows how to provision a site using the SharePoint admin console.
The SharePoint Project Web App allows one to easily share project information. It should be possible for users to update task status using their web browser or an inexpensive desktop client called Project Online Essentials which used to be called Microsoft Project 2016 Lite.