Welcome to the peony forum Ruthanna! Bare root peonies are best planted in the fall like you did with yours. A plant is referred to as a first year peony its first spring/summer in the ground, regardless of how big the division is. The roots are reestablishing, getting used to its new home, growing more "arterial" roots, etc and will be a bit more needy during extreme conditions than it will be when it's been in the ground longer. It's different for peony seedlings—most people count from the first time foliage appears whether you get growth from seed planted the summer before or 3 summers before.
As for last year's stem, herbaceous peonies die all the way back to the ground each fall and no part of that growth above ground is viable in subsequent years. Tree or woody peonies are different—you can and will get new growth on previous years' stems. Itohs are a hybrid between the 2 other types and warmer zones or protected plants can get growth like woody peonies. Both Itohs and woody peonies can also get growth straight from the root like herbaceous but since woody peonies are more often reproduced by grafting stems to herbaceous roots any growth from the ground needs to be monitored to ensure it's the right type.
