Thanks Kelly, your answer resolves my problem and I'll be using that solution.
However, I'm still a little confused as to why this happens.
When I do public searches, i.e. not signed in, on ArcGIS.com, I get the following results:
Search result for feature layer: 'View item details' takes me to the ArcGIS.com 'Overview' page; 'Open in Map viewer' takes me to a new ArcGIS.com map with the feature layer loaded.
Search result for web map: 'View item details' takes me to the ArcGIS.com 'Overview' page; 'Open in Map viewer' takes me to the ArcGIS.com map. From the map, 'Show item details', for the above feature layer, takes me to the ArcGIS.com 'Overview' page.
Search result for web mapping application: 'View item details' takes me to the ArcGIS.com 'Overview' page; however, 'View Application' takes me to the fba.maps.arcgis.com page (the app opens even though I'm not signed in); From the app, 'Show item details', for the above feature layer (from the above map), takes me to the FBA login page.
So the only time that there's a problem is when trying to view the feature layer's item details from the app. Removing the 'fba.maps.' text from the app's url gives me the access I need, which is great.
However, I don't understand why, if the feature layer and web map use ArcGIS.com, the web mapping application uses fba.maps.arcgis.com (while the app's 'Overview' page also uses ArcGIS.com)?
Thanks.