Jennifer, all is good. I am sure (99%) your bud is a TP bud, because it is rather matte. Herbaceous peony buds are usually very shiny.
The growth is coming from the woody stem; that is, the scion, which is a little TP branch grafted on the herbaceous stock. With a little bit of luck, that little bud will form a tiny branch that will get hardy before the fall.
Your TP looks totally OK. No need to worry. I cannot tell if it was planted deep enough, but the new growth looks OK.
It is getting darker outside so I cannot take pictures, but, tomorrow, I'll take a picture of one of my TPs and you will see how they grow.
