Labeling doesn't seem to re-adjust itself on webmap.
For example, while publishing from ArcMap, both labels would show:
But once pulled in as a service on webmap, only the one label shows if the features overlap:
I am not a user of ArcGIS Online - I develop against the ArcGIS JSAPI - but according to the help documentation, it doesn't seem you can do this dynamically:
Create labels—ArcGIS Online Help | ArcGIS
The map viewer places as many labels on the map as possible without overlapping them. Thus, in areas where features are tightly clustered, some features may not get labeled. As you zoom in to an area, more labels will dynamically appear. Labeling priority follows the layer order in your map from top to bottom. Layers at the top get labeled first and thus, will have the most labels. There is no guarantee that you’ll get the labels you want positioned exactly where you want them. Thus, dynamic labeling is best suited for maps where you don’t need precise control and only want to label a few layers.
Thank you, Chris.
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