The word barbaric comes from the Greek word βαρβαρος. It was used initially to describe anyone who is non-Greek in culture and in customs. It's thought that the origins of the word came from the way the Persian language sounded to the Greeks: "bar-bar". As such, βαρβαρος had a negative connotation more or less from the beginning. However, that initial negative connotation was really only to do with speaking--it was mocking the accent with which a foreigner would speak Greek, and did not yet have the connotations that barbarian does in English today (Strab. 14.2.28).

The strong negative connotation is more clearly visible later. It's clear from several sources that the dichotomy implied by use of the word βαρβαρος is with Ἑλλάς (Greece), or Ἕλληνες (the Greeks). This is shown in Thu. 1.3, where Thucydides states clearly that the reason Homer, the oldest surviving Greek poet, does not use the word βαρβαρος is that there was not yet a fully formed Ἑλλάς to contrast it with. A discussion partner of Socrates also clearly states the dichotomy between barbarian and Greek, though he is critical of it, thinking it overly simplistic (Plat. Stat. 262d).

The Persian War was fought between the Greeks and the Persians from 499-449 BC, and is often thought of as the event which crystallized the dichotomy between Ἑλλάς and βαρβαρος, and which established the strongly negative meaning of βαρβαρος. The LSJ, a highly respected Ancient Greek dictionary, defines the word as "barbarous, i.e. non-Greek, foreign...II. after the Persian war, brutal, rude". This is generally correct, though Thucydides, writing in the second half of the 5th century BC, after the Persian war ended, does establish the dichotomy between Greeks and barbarians, as I mentioned above, so the connotations of nationality certainly do linger. Later, Thucydides uses βαρβαρος in the superlative, βαρβαρωτάτους, translated as "most barbarous", which necessarily means that it carries the meaning on an attribute, and not an ethnicity; if βαρβαρος strictly meant non-Greek, then its superlative would be nonsensical, as one is not more non-Greek than someone else (Thuc. 8.98).

It is clear by the time of Demosthenes, a 4th Century BC Athenian orator, that βαρβαρος is used in a negative light, and that it is losing the connotation of ethnicity. In some cases, it seems to have connotations of rashness or impulsivity (Dem. 26 17), and in others with a "hatred of religion" (Dem. 21 150). In both cases, it is used to describe a Greek, though it is not unthinkable that it may carry a connotation of attack on their Greekness.

Later, the word βαρβαρος is used in 2 Maccabees 2.21, written in Greek in the 2nd Century BC, to refer to the Greeks, showing that by that point it had diverged nearly fully from the connotation of non-Greek.

The Greek word βαρβαρος came to Latin as 'barbarus', and then into English as barbarian, and a whole host of other European languages similarly (e.g., German, 'Barbar', Icelandic 'barbari'), completing its etymological journey. The connotations described in the question are mostly intact from over 2000 years ago, passed from Greek, to Latin, and then finally to English.