I have read that Pythagoras's fraternity had two kinds of members, the 'acousmaticians', who were allowed to attend the lectures, and the 'mathematicians', who had been initiated. Is this the origin of the noun 'mathematician' ?
TL;DR: no, it does not seem to be the case that the mathematicians of Pythagoras had the narrowly defined meaning of practitioners of mathematics, but were rather more general scholars.
$\bullet$ Greek: The Greek origin of the word "mathematics" is "that which is learned", and early uses were consistent with this broad sense of the word. A History of Greek Mathematics describes how for Plato μαθηματα
simply means any subject of study; Archytas (428–347 BC) writes that "those concerned with mathemata were particularly interested in the speed of the stars, their risings and settings, and in geometry, arithmetic, astronomy, and also music – for these mathemata seem to be sisters."
The Greek philosopher Sextus Empiricus (160–210 CE) explicitly used μαθηματικούς (mathematicians) to indicate a broad class of scholars, including rhetoricians, geometers, musicians, astrologers.
$\bullet$ Latin: As recorded in the Latin Dictionary of Lewis and Short, early uses of the noun mathematician in Latin had the modern meaning as well as the meaning of astronomer or astrologer, for example, in this famous quote from Augustine of Hippo (fourth century):
Quapropter bono christiano, sive mathematici, sive quilibet impie divinantium, maxime dicentes vera, cavendi sunt, ne consortio daemoniorum animam deceptam, pacto quodam societatis irretiant.
The good Christian should beware of mathematicians. The danger already exists that mathematicians have made a covenant with the devil to darken the spirit and confine man in the bonds of Hell.
$\bullet$ English: The Oxford English Dictionary records earliest known uses of the noun mathematician, which in the 16th century could mean both a person who is skilled in mathematics, as well as an astrologer. Ranulf Higden's Polychronicon is the earliest 15th century source.
In classic Greek, μάθημα is a neuter noun, formed by a standard procedure from the root of the verb μανθάνω, to learn, and denotes in general the object of learning. Also standard is the derivation of the adjective μαθηματικός, "what concerns the object of learning". Plato refers this adjective to persons still in the generic sense of wishful to learn. Aristoteles uses the neuter plural, τὰ μαθηματικά, to denote theoretical studies, including mathematics, mathematical physics, astronomy. As far as I know, Aristoteles is also the first who uses μαθηματικός as a substantive, ὁ μαθηματικός, the mathematician, of course in the wider acceptation of "theoretical scientist" that this word had in all ancient science. In fact, throughout all medieval times, in this sense "Mathematici" especially denoted the followers of Aristotheles in natural sciences, (often as opposed to "Medici", the followers of Hippocrates, scientists with an empiric approach based on observations).
The noun akousmatikos, that however we know from later authors, like Porphyry (born 230 AC) and Iamblichus (born 245AC) has a parallel formation: ἀκούω (to listen) > ἀκουσμα (oral teaching) > ἀκουσματικός (listener, follower of oral lectures). The distinction is between advanced students and not initiated, simple listeners, and also emanated the two branches after the scission of the Pythagorean school: one more interested in scientific aspects of Pythagorism, the other attached to the religious aspects of it.