Terms such as single, lotus-shaped, crown and double describe the structure of a peony flower. They are helpful shorthand, but they are not a substitute for a cultivar name.
Look at three things in an open bloom: whether the centre is visible, how the outer petals sit, and whether the inner petals form a distinct raised group. These are easier to compare than trying to count petals on a moving flower.
Use more than one photograph
An opening bloom can look different from the same flower several days later. Ask for a fully open example and, if possible, a side view. A photograph looking straight down can hide a raised centre or make a shallow flower look more rounded.
Do not use flower form to decide whether the plant is a tree peony or a herbaceous peony. That is a separate question about the plant's growth and stems.
For a purchase, record both the named cultivar and the form shown by the nursery. If the name is missing, you may be able to describe the flower, but you do not yet have a reliable identity for the plant.
Related reading: Peony Flower Forms: Single, Lotus, Crown and More.