Gábor Szabó One thing to note is that the new service will also be integrated into the Office365 Suite. https://project.microsoft.com/en-US/ is the new Home page - so you are in fact looking at the correct, new experience already.
The question of when to move to the new system is a case by case basis based on requirements. For highly mature PMOs, the existing PWA functionality might be more appropriate for a while as it will have richer portfolio management, but this maturity is fairly rare.
As for the name, we are simplifying all of our naming as more and more of our customers move to the cloud. it did not make sense to call it Project Online when all of our customers are moving online.
The new service saves projects into the Common Data Service for Apps (or CDS for short). SharePoint will remain a critical part of the service as the place to store documents and collaborate because Project will be integrated with Modern Groups.