That's correct. The options you get to tailor how an existing web map is displayed inside a Cascade story are limited to things that don't make changes to the web map. So you can choose the extent that is displayed, turn layers on and off, and also choose to display the popup for a particular feature. These can be applied to any web map, not just maps that you own. This is a nice feature of Story Maps because you can use maps created by your colleagues, or anyone in the ArcGIS community, not just maps that you author. This applies to all the Story Map apps except for Map Tour and Shortlist, which in most (but not all) authoring workflows expect you to own the web map in which the places you display are stored.
To get that additional look for your map, you could do a Save As like you describe. But you could also just have two layers in your map showing the same data with different symbology, and then turn one of those layers off when you add it into the respective view in your Cascade. Keeping everything in one map can be easier to manage, ensure that everything is consistent and can also improve drawing performance, because the Cascade app just needs to load one map. But this may depend on how else you may be using the map: You may want to use that map in another app in which you wouldn't want to two different layers for the same data to appear. But in general, we'd recommend sticking with one web map if you can.
Rupert
