It's obviously not important in itself whether class names end in Manager or Service. What's important, generally speaking, is that names accurately convey what is being modeled. And that's the crux of the problem: "Services" or "Managers" aren't real-world objects that we try model in our software objects. Rather, they are places where we collect a bunch of methods that do stuff that simply doesn't fit in with the responsibilities of any of the objects we do need/want to model.
Personally I prefer "service", but only because "manager" seems like something one could actually model, i.e. there could be real-world managers that our "-manager" objects represent. But the point is entirely academical and I immediately concede that it makes no practical difference whatsoever.
What really matters is usually far more basic than such fine points: To have a model that is well understood by all involved in development. If my experience is anything to go by, that is seldom the case. My tip to those asking if "manager" or "service" is the right metaphor is therefore: Flip a coin, make sure everyone knows about the convention, and spend your time pondering and discussing matters that matter!