So are saying the add-on card works with the motherboard's built SATA controller if there already was a SATA system setup and SATA Controller is an independent system that works with the motherboard without any SATA system built in?
Yes, the addon is using the motherboard SATA controller(it is connected to sata ports on the motherboard), while a SATA controller is independant and uses the pci bus power. Its is not connected to sata prots on the motherboard like the add on.
Both cards you pointed are controllers.
Hers an example of an add-on sata:http://www.newegg.com/Product/Product.aspx?Item=N82E16812206003