Do the Taylor Brethren use the term 'Trinity' in a specialised way?

I found an archive on-line, with a section on The Godhead and Sonship, https://www.brethrenarchive.org/archive/later-exclusivism/raven-section/taylorites/eternal-sonship/the-godhead-and-sonship-a-reply/

The main heading was, “New Teaching on Divine Relationships 1929” The documents listed were not accessible to me, but the first page of the following document was readable. It had to do with claimed error of Mr. Taylor who

“…has ventured upon ground beyond his rights, denying the eternal relationships of Father and Son, the Son’s eternal home in the Father’s bosom, and His eternal distinction as the Logos. This has called forth written papers from many able pens, and his attempts to justify the error shew a departure from sound doctrine which may be made plain to the simplest.

The first and main point to observe is the failure of the writer to distinguish between the ontologic and the economic, or, in other words, his constant confusing essential unrevealed Deity with the revelation of God in tri-unity as Father, Son and Spirit.

Apprehending this it is easy to see how he shuts out from the saints the revelation of eternal love, light and glory, made known to us by the Son, as well as the anticipatory activities of Father, Son, and Spirit in that eternal past, in an endeavour to guard that which no one but himself is encroaching upon, namely, essential unrevealed Godhead. This is the secret of the whole defect…” Relations in the Godhead as Revealed in the Counsels of Eternity. Thoughts on a book entitled: ‘Names of Divine Persons’, by James McBroom

That is just the introduction to James McBroom’s booklet, and he indicates that he will set forth his proofs.

There were several other documents listed. I noted that one was called, “The Truth as to The Trinity – with special reference to a pamphlet by C.A.C.” [= C. A. Coates] by R. E. Elliott, 31 pp, circa 1930. Also:

“A Letter on Eternal Sonship with Notes by W. H. Westcott” 18/8/31, 19 pp. Another paper by W. H. Westcott was dated 20/3/1932, 4 pp re. John 1:18. If W. H. Westcott’s views reflect anything of Unitarianism, that would be significant, but I cannot access his letters on that site. Other documents are:

“The Godhead and Sonship – a reply” by Frank Wallace, 10 pp.

“Reversal not Adjustment – an appeal to my brethren” by Ernest Middleton. An insider writing against the new teaching on the ‘Eternal Sonship’, 24/10/1932, 24 pp.

“The Eternal Sonship – a study in scripture” by C. F. Hogg, an ‘Open Brother’, following the controversy caused by the London party’s departure from orthodoxy. November 1932, 32 pp.

“Reply to ‘An Open Letter’ by A. J. Pollock” by D. L. Higgins circa 1933, 8 pp.

“The Divine Sonship of the Lord Jesus Christ” by V. W. J. H. Lawrence, 1933. “Recovery, An Historical Review” by D. L. Higgins circa 1935, 15 pp.

Someone had posted this comment on 24 March 2021: “George Davidson (under the pseudonym of Quartus) also penned a helpful tract entitled “The Son” in reply to C.A.C. [= C. A. Coates]. It can be found on STEM here:

https://www.stempublishing.com/authors/davison/QUARTUS.html

Some points made therein indicate this to be a short criticism of a tract called "Remarks", which apparently explains the theory of "Non-eternal Sonship". Mr. Davidson cannot agree with that view. The whole of his criticism needs to be read, but here are a couple of points he makes about contradictions made by the author of the tract, "Remarks" (C. A. Coates, I gather):

If we have got the thought our author [C. A. Coates] is trying to establish, we think he is trying to make out that there is a distinction between the Person Who is eternal and the Son — which he asserts, is only true of our Lord in Manhood. Simple persons like you and me [sic] may be excused if we cannot see this fine distinction in Scripture. But we need not be dismayed as it is evident our author is not clear himself. More than once in this tract he speaks of the Person of the Son. He has told us that in Person He is Son. He has also said — His Person is eternal. Yet on these two pages [13 & 14] he is labouring to try and prove a difference between the eternal Person and the Son.

Davidson has much more to say about this, but another important reason for his criticism of that "Remarks" tract follows a little later, regarding page 33:

In reply to A.J.P. concerning the meaning of the "Only-begotten Son," C.A.C. is almost guilty of a deliberate deception. Note how in reply to such verses as John 1:18, C.A.C. quotes Heb. 1:5. Now why did not C.A.C. tell his readers that Mr. Pollock, was speaking of one word and he another? No one that I know will doubt that Hebrews speaks of our Lord as coming into the world in Manhood, but, to put this against what A.J.P. was bringing forward, shews either great ignorance of words, or an attempt to deceive simple readers. If you have the means, look up these two words and you will find that the word A.J.P. is using is — MONOGENES. The word C.A.C. is using is — GEGENNEKA [GENNAO]. The first means, as it is translated — "Only begotten," the second — "Have begotten." Why then does our author try to nullify the meaning of one word by adducing another with a totally distinct meaning? Does C.A.C. think he is going to get students of Scripture to accept such preposterous conclusions? To think that because gegenneka [gennao] means, "have I begotten thee," we are to accept that monogenes "Only-begotten" means the same, is surely so absurd, that no right thinking person taught of God will accept it for one moment. It was might do for the simple members of the London party who are prepared to say yes to these monstrosities, but no one else I am sure.

The point at issue is, whether the word "Only-begotten" refers to our Lord before coming into Manhood, or is only a time name given to Him when here. This is why he puts forward Heb. 1-5, where coming into the world is the simple meaning. It is a bad omen to note how he presses for the literal meaning of "begotten," when A.J.P. says "Scripture forbids its being used in the case of the Son of God in the sense of derivation, or generation." The fact that Scripture does use words that can only be understood by their Scriptural use, is wholly lost sight of. He knows, as well as you and I know, that to take up many words in Scripture and give them merely their English meaning, would make even God a time Being. No, we are not going to accept that in John 1:18 "Only-begotten" means begotten as we understand it in grammar. Here is a note by T.H.R..

"the Greek word for only-begotten is the Septuagint translation of only-one; in Hebrew, sometimes "Beloved," sometimes "only-begotten," is used in the L.X.X. for the same Hebrew word signifying "Only-one." In English it is translated "Only." Gen. 22. "Thine only son whom thou lovest." In Ps. 22 it is the same word "My Darling;" the margin puts it "Mine only-one." That is really the meaning of it. Only-begotten has nothing to do with being born at all. It is simply the fact that He is God's only beloved Son.

Again, a great deal more follows to explain this, and other errors in the tract under examination. I can only gather that there has been some departure from orthodox use of the term 'Trinity' by some of the Brethren, but my sources are from the 1930s era and I have not found anything current, or from the Australian Brethren. I am not in a position to directly answer either of your two questions, but there are definitely grounds for asking those questions, and reason to suppose that the answer to both of them could well be, "Yes".