Random House unabridged:
Nestorian - one of a sect of Christians, followers of Nestorius, who denied the hypostatic union and were represented as maintaining the existence of two distinct persons in Christ.
Nestorius - died A.D. 451?, Syrian ecclesiastic: patriarch of Constantinople 428-431.
Syriac - a form of Aramaic used by various Eastern Churches.
Chaldean - 1. one of an ancient Semitic people that formed the dominant element in Babylonia. 2. the indigenous Semitic language of the Chaldeans, Aramaic being used as an auxiliary language.
Nestorianism - [Theology] the doctrine that there were two separate persons, one human and one divine, in the incarnate Christ. It is named after Nestorius, patriarch of Constantinople (423-31), and was maintained by some ancient churches of the Middle East. A small Nestorian church still exists in Iraq.
Chaldean - (n.) 1: a member of an ancient people who lived in Chaldea c. 800 BC and ruled Babylonia 625-539 BC. They were renowned as astronomers and astrologers.
2: the Semitic language of the ancient Chaldeans.
• a language related to Aramaic and spoken in parts of Iraq.
3: a member of a Syrian Uniate (formerly Nestorian) Church based mainly in Iran and Iraq.
(adj.) 1: of or relating to ancient Chaldea or its people or language. ...
2: of or relating to the East Syrian Uniate Church.
Assyrian - (n.) 1: an inhabitant of ancient Assyria.
2: the language of ancient Assyria, a dialect of Akkadian.
3: a dialect of Aramaic still spoken by a group of people of mainly Christian faith living in the mountains of Syria, northern Iraq, and surrounding regions.
(adj.) 1: of or relating to ancient Assyria or its language.
2: relating to or denoting modern Assyrian or its speakers.
Uniate - (also Uniat) - denoting or relating to any community of Christians in eastern Europe or the Near East that acknowledges papal supremacy but retains its own liturgy: the Uniate churches.
New Westminster Dict. of Liturgy and Worship, ed. Davies:
Nestorian Church - see East Syrian Worship
East Syrian Worship - 1. Historical. The East Syrian Church (known to many as the Nestorian or Chaldean church) is the Syrian Church of Antioch as it developed east of the frontiers of the Roman-Byzantine empire. Its centre was Nisibis ... During the seventh and later centuries this church spread to Turkestan ... with bishops in Samarkand, Tashkent, Karakoram and also in Tibet as well as in China and India. Today this church, except a part now in the Roman Catholic communion, is limited to small pockets in [the] USA, Iraq, Iran and India. ...
2. Liturgical books. The main eucharistic liturgies are three, which go by the names of (a) Addai and Mari, (b) Theodore of Mopsuestia, and (c) Nestorius ...
3. The eucharistic liturgy. What has astonished many liturgists about the liturgy of Addai and Mari is the absence of the words of institution; this is not unusual in the West Syrian tradition either. ... This is shocking only to those who believe that the recital of the words of institution effect[s] the consecration. ...
West Syrian Worship - The West Syrian Church, known to many as 'Jacobite' (after Jacob Baradeus, the sixth-century reorganizer ...) and as Monophysite (after the erroneous idea prevailing in Byzantium and the Latin West that the West Syrians believed only in the divine nature of Christ), historically inherited the Semitic, Palestinian tradition of Christianity, though not uninfluenced by the Hellenistic milieu in which they lived.
The Syrian tradition broke up soon into four families – the East Syrian (Edessa), the West Syrian (Antioch), the Melchite (Greek), and the the Maronite (Lebanon). ...
The liturgy is nowadays celebrated mostly in the vernacular – Arabic in the Middle East, English in America, Malayalam in India, and so on – though certain portions may still be said by the priest in Syriac.
w, I had never even heard of them, but here's a little of what my reference sources have to say about them. (You're welcome.)
I assume you're well aware that Nestorius was a person, that there's no such language as Nestorian, and that that has nothing to do with the difference between Syriac (the language) and Syrian (the region or tradition).
Chaldean, however, is evidently both the adjective for the historic geographic region of Chaldea and the noun for its language. (Random House also lists an old form for the language, Chaldaic, that seems to have fallen out of use.) And Assyrian in the modern sense is evidently either the language or the people who speak that language. Very confusing.
Of course you can use Syrian alone whenever it's already clear from context that you're talking about religion and not about the country Syria. But for a dictionary listing in this sense, I wouldn't use Syrian without another word like Orthodox, Church, Christian, etc.
In English at least, the formal/official name of at least one such church now seems to be the (East?) Syrian Uniate Church, rather than Syrian Orthodox or whatever. If you need to know more, you could look that up; it may be the part that has reconciled with the Roman Catholic Church, but that's just a guess.