Interesting)
No, we haven't such words, but we use another way.
My full name is Aleksander. But my friends or family never use the full name they call me Sasha - short name. Or invent something new)
When I was a boy everyone could call me by short name.
Nowdays it depends. My chef or some older person could call me by short name or by full name, whatever they want. Usually be full name - Aleksandr
But I will call them by full name + patronymic. For example my full full name is Demyanov Aleksandr Petrovich. When I would be old, people would call me Aleksandr Petrovich.
P.S. Petrovich is because my father's name is Petr, patronymic formed from father's name. My daughter is Anna Aleksandrovna Demyanova.