Hi there, 

I can see you are creating fme_line geometry. This can be discovered under fme_geometry in Feature Information window. 

I inspected the 9 features on the Sampler

To write these 9 feature to Esri geo database, you have two choices

1. go to Writer > Add Writer specify the Esri format, the name of the object class (feature type) and the dataset path to where you want it saved. Then be sure to make the     geometry = %_line from the dropdowns. The user attributes are typically set to Automatic, but you can change to manual and alter the schema to match a standard or existing table
2. click on the sampler 9 features and in the canvas, type in Feature Writer. A workflow transformer will be place and you essentially so the same as steps above in 1.

Like I said in one of my answers to the original question, maybe you don't set the geometry type to geodb_polyline.

I exported to 'ESRI Geodatabase (File Geodb Open API)’ and set the Geometry type to geodb_polyline. When I open this geodatabase-folder with FME Data Inspector (I don't use ArcMap) I see all the features I exported, with their attributes.

Is there and difference in table handling...settings 

Sorry, I can't help you there.

I can export to 'ESRI Geodatabase (File Geodb Open API)’  only, and that works as expected, but I have no access to the other ESRI geodatabase formats.

But as that export works correctly (at least I can open the resulting gdb and display the data), I would expect the problem to exist somewhere else.