I hesitated to use some different material names for each part of the handrails and balusters, but for U shaped staircases, it would make many materials with the risk of confusion for beginners. Maybe, I could add an option for a limited list or a full list of materials.

Better a long list than not having the option. That's why I did this with the public staircases I published. With those you can even differentiate between the first and second landing and choose two different baluster types (panels or glass). This is important for the different positions where the staircase is placed against a wall or not. Sure, it's a long list of options but you don't see that in the plan, only when modifying the textures. It's not about being able to set a different color or texture, it's required to allow making a part invisible, that's important.
[hr]I agree with the name changes sjb007 proposed. Specifically because generally "Stairs" is considered the complete staircase and not a part of it. The proposed names are less confusing.
Here's a diagram with (US) part names for a staircase: https://www.stair-parts.com/parts-of-a-staircase-diagram-glossary-list-of-stair-part-names/
Landings can also be called platform.
[hr]Could you share how to set randomized coordinates for the textures as you did for the threads and risers? I wanted to do that for the public staircases I created but didn't want to have a property for each separate thread and riser. That would make the material list a 'little' too long (although it would allow to create rainbow steps [;)] )