Or was anything carrying sound to remote already called phone before telephone?
Lover's String (also found as Lovers' String or Lover's Telegraph) was the name used in the late 19th Century in documents relating to the development and patenting of the electric telephone.
In documents relating to Mr Bell's 2nd patent (#186,787 of Jan 30, 1877), there is discussion of the origin of the telephone in which we find this text:
He shows and describes it as a "stretched" diaphragm, with a piece of soft iron attached to it, showing that it is to be an elastic membrane, and not of iron. He got his idea from the lover's string telephone, which is ordinarily made with a diaphragm of parchment or bladder.
Conklin's Handy Manual of Useful Information and World's Atlas of 1891 (as well as several other publications) mentions this:
In 1831, Wheatstone showed that when the sounding boards of two musical instruments were connected together by a rod of pine wood, a tune played on one will be faithfully reproduced by the other. Somewhat later a toy, called the Lovers' String, was made, and is the simplest form of a mechanical telephone. The toy consisted of two tin cups, the bottoms made of parchment or cat gut tightly stretched like a drum head, and connected, one with the other, by a string or cord. When the string was drawn taut, sounds such as those of ordinary speech produced in front of one of the cups were transmitted along the string to the other cup and reproduced there. This was the first telephone.
There is an image of the text in a different publication, here.
According to the book, The Telephone and its Many Inventors (Google books link), the device you mention was first attributed to Robert Hooke in 1664. Apparently, he did not give the device a name:
I can assure the reader that I have, by the help of a distended wire, propagated the sound to a very considerable distance in an instant, or with as seemingly quick a motion as that of light, at leas, incomparably swifter than that, which at the same time was propagated through the air; and this not only in a straight line, or direct, but in one bended in many angles.
According to this site, it might have been simply called an instrument or a communication system, but I could not find anything definitive. As a now deleted answer stated, the word telephone did not come along until 1835.
According to the Wikipedia article on the "History of the Telephone", the "two tins cans linked by a string" phone is formally called an "acoustic telephone", and "has been known for centuries".
Also from Wikipedia, in its article titled "Telephone", it is noted that the Greek-derived name "telephone" had been used for a few other inventions prior to the electric telephone we know today, including a system of blaring horns used to communicate between ships at sea.