Both of them are Civilian clothing-buttons. The 2-piece type you found date from the 1830s and onward. We know that some Confederate (and yankee) soldiers did use these Civilian buttons on their uniforms, but only when they couldn't get the correct Military buttons to replace one lost from their uniform. That is why these Civilian buttons are found in civil war army camps ..and it's why they often get incorrectly advertized on Ebay as being "Confederate buttons." But they were not manufactured for Military use. Even today, a Soldiers' motto is "When in need, you use whatever you can get."
These Civilian buttons, sometimes loosely called "flower buttons," were often gold-gilted. If you (gently) clean them, you're probably see some gold on their fronts.