All of the above is correct. However, the main reason that Italian became the international language for musical performance directions, over all other languages is remarkably simple. In the 17th, 18th and early 19th centuries, many people could not read or write in their native language, let alone in any other. Also, the majority of workers were labourers who had to work very long hours, just to stay alive and support their families. The only people seriously interested in the Arts & Sciences were the wealthy, educated sections of society who had the money and leisure time to indulge these interests.
The children of these families had private education, and in Western Europe, particularly, the wealthy made sure that their children people were taught science, the arts and languages. Latin and Greek were often taught, along with modern foreign European languages and customs. The mortar which cemented what was taught, was to arrange for the newly educated young people to travel, and experience what they had learned, in foreign countries, and by interacting locally, with other cultures. These young people would have no issue with 'allegro', 'dolce', or any other Italian words, as they would already have a good idea of the language of one of the greatest countries of the time, and its considerable influence in religion and the arts and sciences throughout Europe. They would see no need to translate these words, as they were already a part of their general education. They would no more think of translating these words than we would, of translating, 'fiancé', 'crescendo' or 'zeitgeist'.
It was not until many years after the French Revolution that serious nationalism began to take root, and composers started to use their own languages for performance directions. So, as has already been mentioned above, we get the somewhat amusing situation of a composer such as Debussy (French) or Beethoven (German) sprinkling performance instructions in their own language, whilst still employing Italian for much of the conventional usage! As Nationalism grew through the 19th century, so more and more composers began to use their mother tongue to describe HOW the music was meant to be performed (quickly, slowly, majestically, sadly, etc) and that is how and why the situation developed from there to the present day.
