Most of the moves with names in ice dancing are specific steps and turns that the skaters do with their feet. You might not be able to tell the difference between them if you're looking at the top of the body. The judges and technical specialists are looking very carefully at the feet, but fans usually prefer to watch the whole body.
There are also names for different kinds of dance holds
Lifts are named by the pattern that they make on the ice: for example, "rotational lift" or "curve lift" or "straight-line lift." You can tell if they're rotating or curving or moving in a straight line during the lift.
Different teams will have different positions of the man and the lady in their lifts, but the positions don't have official names. Sometimes teams make up brand new positions that no one ever used before, so no one ever had a chance to name it before. The teams might make up their own names for a specific lift that they're doing that year, so they can talk about it to each other while they're practicing.
In general, the technical parts of ice dancing tend to have formal names. The creative parts usually do not have formal names.