Ask Your Users
That’s really a question for your users, not us. In the course of your user research, observe how the users refer to servers. What language do they spontaneously use? If all else fails, ask them questions like, “If you heard that the app autosaves your documents to the server, what would that mean to you?”
Rather than asking us to check up on popular websites, do it yourself, favoring the sites you know your users use (e.g., the legacy app). Or, if your users are just a general population of internet users, you can type into Google text like, “Autosave files to,” (or whatever feature we’re talking about) and see what autocompletes. That’ll tell you the most common terms for that context. You can also type in full phrases of alternatives to see what has more hits. “Server” wins over “Cloud” for autosave and synchronize, but “Cloud” wins over “Server” for backup, so it depends on the exact feature you’re talking about. I also get brand names like Azure, Onedrive, and iCloud, which suggests that if your app is using a heavily promoted on-line service, then you can use it by name.
Function not Implementation
That said, most the time, you don’t want to talk about technical implementation at all. Users probably don’t care if it’s a server or cloud or session. They just want to know what they can and can’t do, and what features are functioning. For example, rather than saying “Disconnected from our server,” say “Autosave temporarily unavailable” (assuming your users use words like “autosave”).