What does it mean to say the Bible is the "Word of God" and how is it justified? How has the phrase the "Word of God" been understood (as applied to The Bible) in the Christian tradition? I assume different sects of Christianity answer this differently and I'm interested in hearing all answers, specifically with regard to the doctrine of the infallibility of The Bible.

I'm familiar with 2 Timothy 3:16: 


  "All Scripture is inspired by God and profitable for teaching, for
  reproof, for correction, for training in righteousness..."


However, what is the definition of "Scripture" and why are the letters of the New Testament considered scripture?
Karl Barth believed that Jesus is the Word of God, but that the written Bible is NOT the Word of God. He believed that the Bible becomes the Word of God when preached by the church. See https://postbarthian.com/2018/08/20/karl-barths-doctrine-of-the-word-of-god-in-threefold-form/


  Barth then argues the Bible becomes the Word of God when it is
  proclaimed by the living human voice of the Church, and only in that
  event may the Bible be rightly called the Word of God. Barth explained
  "the Bible, then, becomes God’s Word in this event, and in the
  statement that the Bible is God’s Word the little word 'is' refers to
  its being in this becoming. It does not become God’s Word because we
  accord it faith but in the fact that it becomes revelation to us."


On the other hand, Fundamentalists, Evangelicals, and even Catholic traditions going back to Augustine subscribe to various theories of inerrancy and canon formation. This article has some intersting points to make about where the ideas of inerrancy come from:

https://www.thegospelcoalition.org/article/did-fundamentalists-invent-inerrancy/

Many people rely on their view as handed down to them by their church statement of faith, denominational position, or an ecumenical creed. One such creed is the Chicago Statement on Biblical Inerrancy:

http://www.romans45.org/creeds/chicago.htm

Here is the summary statement of the much longer document:

I. SUMMARY STATEMENT


God, who is Himself Truth and speaks truth only, has inspired Holy Scripture in order thereby to reveal Himself to lost mankind through Jesus Christ as Creator and Lord, Redeemer and Judge. Holy Scripture is God's witness to Himself.
Holy Scripture, being God's own Word, written by men prepared and superintended by His Spirit, is of infallible divine authority in
all matters upon which it touches: It is to be believed, as God's
instruction, in all that it affirms; obeyed, as God's command, in all
that it requires; embraced, as God's pledge, in all that it promises.
The Holy Spirit, Scripture's divine Author, both authenticates it to us by His inward witness and opens our minds to understand its meaning.
Being wholly and verbally God-given, Scripture is without error or fault in all its teaching, no less in what it states about
God's acts in creation, about the events of world history, and about
its own literary origins under God, than in its witness to God's
saving grace in individual lives.
The authority of Scripture is inescapably impaired if this total divine inerrancy is in any way limited of disregarded, or made
relative to a view of truth contrary to the Bible's own; and such
lapses bring serious loss to both the individual and the Church.


Some people rely on certain Bible passages to justify their beliefs about what the Bible is. Thessalonians equates the Scriptures with God's universal, authoritative, and powerful truth:


  13 And we also thank God continually because, when you received the
  word of God, which you heard from us, you accepted it not as a human
  word, but as it actually is, the word of God, which is indeed at work
  in you who believe. (1 Thessalonians 2:13)


Peter's words are used to establish that Paul's letters are part of Scripture:


  16 He writes the same way in all his letters, speaking in them of
  these matters. His letters contain some things that are hard to
  understand, which ignorant and unstable people distort, as they do the
  other Scriptures, to their own destruction. (2 Peter 3:16)


Similar bootstrapping techniques are used to tie in the other New Testament letters and books. As for the supernatural power and efficacy of the Word of God:


  12 For the word of God is alive and active. Sharper than any
  double-edged sword, it penetrates even to dividing soul and spirit,
  joints and marrow; it judges the thoughts and attitudes of the heart.
  (Hebrews 4:12)


On the other end of the spectrum, you have people whose justification for the Bible being the Word of God is because it is self-authenticating. The Holy Spirit applies the words of the Bible to a person's life and they are transformed. Their personal experience gives witness to their belief. I began as person who accepted it as told to me by people I trusted (creedal) and ended up as a person who experienced the power of the Bible to change my life. For example, my Bible Study leader in college asked us all to memorize Galatians 2:20-21:


  20 I have been crucified with Christ and I no longer live, but Christ
  lives in me. The life I now live in the body, I live by faith in the
  Son of God, who loved me and gave himself for me. 21 I do not set
  aside the grace of God, for if righteousness could be gained through
  the law, Christ died for nothing!”


A month later, my fear of death went away. In like fashion, I attended a retreat to study Philippians, and Joy entered my heart and overcame a decade of depression. A passage in Isaiah and a Psalm stopped the nightmares I used to have all the time. The Sermon on the Mount helped me to not feel poor and incapable of helping others. I could go on and on. Ultimately, it is the believer's personal experience of deliverance and growth that proves that the Bible is the Word of God.