There certainly are several "takes", or, "variations" on the Trinity doctrine, but there is only one orthodox Trinity doctrine that Catholicism, Orthodoxy, and Protestantism adhere to. Those with different "takes" on it, are all outside of what is commonly viewed as orthodox Christianity. That point is shown in the question you gave us a link to, in your comments.
The question was asking about Latter Day Saint interpretation of the Trinity doctrine. A comment from the OP was, "Scoping the question to Trinitarians in general may open the door to doctrines that may have some biblical basis, but are mostly based on tradition". This was a reference to the LDS "variation" on the Trinity doctrine, which is so far removed from the orthodox one as to render the LDS faith outside of Catholicism, Orthodoxy and/or Protestantism.
There are other variations that call into question whether the people concerned can fit the orthodox definition of adhering to Christian doctrine in this regard. Some claim they believe in the Trinity, yet say Jesus is "a God", or "divine" which repudiates the orthodox doctrine.
That is why the OP in the question you linked us to wanted answers from those who are in the "Sola Scriptura" camp. They stick purely to what the Bible says on the matter, avoiding reference to tradition as proof, or much later claims to angelic revelations, or visions / information from God or Jesus. But it needs to be pointed out that when it comes to the Trinity doctrine, Catholicism, Orthodoxy and Protestantism are agreed on the biblical basis for it. Without any appeal to tradition, the doctrine stands on what the Bible states. However, the LDS religion depends on early 19th century revelations and changes to the Bible by its founder (Joseph Smith) to come up with their Jesus being created by a God called Elohim, having union with his spirit goddess wife, their firstborn spirit baby being called Jehovah, who later became the man Jesus.
This is a classic example of what you ask for: a "doctrine based on tradition, and scripture offers secondary support only". Orthodox trinitarianism categorically states that there was never a time when "he was not" and that all who say the Christ came into existence at a certain point in time are anathema - cursed.
The LDS religion cannot claim to believe in that Trinity doctrine, wherein Father, Son, and Holy Spirit are all uncreated and eternal. The LDS variation on that doctrine requires its early 19th century traditions and interpretations of its founder.
Further to this: "Is there a sector of Trinitarianism whose doctrinal basis is tradition first and scripture second?" This is - to orthodoxly trinitarian believers - a bit of a trick question (though you won't realise that). The point is that no appeal to scripture even as a secondary support, can square with a doctrine that has a created Christ. So, even if (for example) it is claimed that the LDS idea has its tradition first, with scripture supporting that in a secondary way, that could not be true. The only orthodoxly trinitarian doctrine rejects any created status to any of the three triune members.
Because of persistent difficulty protecting the Trinity doctrine from those who wanted to slip in the idea that the Son of God was created, creeds were formed. The Nicene Creed expressed belief in the one Lord Jesus Christ, "Begotten of the Father before all the ages, Light of Light, true God of true God, begotten not made, of one substance with the Father... And in the Holy Spirit... who with the Father and Son is worshipped together and glorified together..." [emphasis mine]
The Creed of Nicaea (325) was produced to exclude Arian error. It had to employ technical language and solemn curses (anathemas) as Arius seemed to be able to agree to any statement using solely biblical language. That is why the Creed of Nicea states (regarding this) starting similarly to the Nicene Creed but adding:
"And in one Lord Jesus Christ, the Son of God, begotten of the Father,
only-begotten, that is, from the substance of the Father; God from
God, Light from Light, Very God from Very God, begotten not made, of
one substance with the Father... And those who say 'There was a time
when he was not', and: 'Before he was begotten he was not', and 'He
came into being from nothing', or those who pretend that the Son of
God is 'Of another substance or essence or 'created' or 'alterable' or
'mutable', the catholic and apostolic church places under a curse."
The Athanasian Creed (late 5th century) was still doing battle with attempts to change the Trinity doctrine, as this book explains. It
"...is directed against the 'modalism' (similar to Sabellianism) which
Priscillianism had revived in Gaul and Spain in the fourth and
following centuries, and against the Arianism of the Goths and
Vandals, which made the Son and the Spirit into second- and third-rank
divinities." (The History of Christianity, p169, Lion 1977)
And this battle continues to this day. There are outright deniers of the Trinity doctrine (so there is no problem seeing where they stand in their opposition to it), but others try to give the impression they believe the Trinity doctrine when, in actual fact, they deny a core, foundational part of it, that the Son was never created. They have a Christ who had a starting point in time. They do not believe the orthodox Trinity doctrine at all, despite their claims.
This means that the answer to your main question, "Are there different Trinitarian foundations?" is that there is only one orthodoxly Christian, trinitarian doctrine. All other presentations of it are not founded on the orthodox Trinity doctrine at all.