And the capital dialect is the predecessor of Putonghua, the modern standard colloquial Chinese.

According to my knowledge on history, "Mandarin" was from Portuguese and its origin is exactly your explanation above. In the 16th century when Portuguese navigators arrived the "East Indies"(East Asia and South&SE Asia), they used this term to refer to any high-ranked local officials. But as the Portuguese got closer involvement into the Ming Empire(China), they gradually narrowed down its meaning by referring to only Chinese officials. In ancient times, high-ranked Chinese officials were forbidden to have their career in the vicinity of their hometowns and they worked in a alien land speaking different dialects. So generally, the high-ranked Chinese officials had to learn the dialect spoken in the capital, the lingua franca at that time, in order to communicate with the emperors, other officials and their subjects. When Portuguese met the high-ranked Chinese officials in South China ports, such as Macao or Canton(Guangzhou), these officials speak the capital dialect different from Cantonese. Therefore, the foreigners called the officials "Mandarin" and the capital dialect " Mandarin". This tradition was inherited by the English-speaking navigators in the following centuries.