By default, this widget pulls in the group listed in the basemap gallery of the organization settings as the source. To change the order you would need to create a group of the maps you would like to include at the organization level. I am guessing you would like the ability to have the option to configure your own basemaps for this widget without having to create a separate group of basemaps in your organization. Is this correct?
That looks pretty close to what I'd like to do but we have a lot of content in my org and I just want the basemaps customized for this one specific publicly accessible Experience Builder app. I just want people to be able to toggle between the streets and the imagery in the app and have the rest of the content in my org unaffected. If there's a way to do that with groups, I'm open to it but I'm not seeing it.
Ok, then this won't work because it will affect all others in your org. You could basically have two web maps, one with the streets and the other with imagery and it will have that toggle in the left-hand corner for the map widget. Since they share the same data source if you have other widgets like the list or any other widgets that use this feature layer you could just add that as a data source and configure it through there.
Another option if you're using the developer edition is you could import the Basemap toggle widget into a custom widget to add the toggle to switch between the two base maps.
Related to this topic, what behavior is meant to happen for experiences shared with the public? When I am signed in and there is an active token I am seeing my organizations basemap and search widget configuration, however in incognito mode I am getting ESRI basemaps and Esri Geocoder.
My experience is hosted in arcgis.com, so I am not using Dev Edition.
Our Organizational Basemap and geocoder items are all public facing, but unlike WAB I am not able to configure it successfully to consume. Any advice welcome.