Ian, here's what I have figured out so far. I have used ArcGIS Online for a while. I guess you kind of figure out how it works by trying different things. The help is not great.
I think the templates you download are for doing a version of online maps called Storymaps? I have dabbled a little with it. Those are for hosting on your own web location and use maps/content generated on a ArcGIS Online map. The templates will have a readme file that you can open and look at, it helps a little. But you have to know some HTML coding to really make the Storymaps work well.
There are a couple templates that host directly on ArcGIS Online and that have a config editor to help you do some very basic tings.
As for your ArcGIS Server, that is useful for serving out your own data layers and incorporating them into your ArcGIS Online maps. But there is also a lot of data to tie to on existing servers out there, but you have to search a lot of it out yourself.
Hope that helps some.