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a)Is a unit of TWC a unit of TWC?
b)Is a unit of TWC with an attached IC not a unit of TWC?

Do the rules prevent multiple types of IC's joining the same unit?
The rule reads as the units you can join, neither the contents of the units you can join nor the limited non IC contrainin gunits you can join.
What you need to show here is that the rules are more specific then saying anything more then TW lords can only join these two cavalry units. I don't read any such depth to them.
The nature of a unit doesn't change from the IC's that join it, a unit of scouts are still scounts and a TWC with two lords attached is still TWC.