Silent call is still the general term that you are looking for.
Nowadays, it is usually initiated by companies by using technology but it can be deliberately done by a person as well.
A silent call happens when your phone rings but there's only silence when you answer it and you have no means of knowing whether anyone is at the end of the line.
For many people, silent calls can be unnerving or indeed frightening. Almost always there is nothing sinister going on. They are usually caused by companies using predictive diallers or other automated calling systems to make large volumes of calls.
It is also mentioned as a type of nuisance calls:
Nuisance calls are calls that are not indecent, threatening or offensive, but which cause annoyance or anxiety. Silent calls or calls at unsociable hours are two examples.
I even found an example that covers faint breathing part in a silent call: (from the book Preacher Man By Charles Carroll Lee)