The doctrine of Sola Scriptura is generally defined as the belief that, while tradition, reason, and experience may be sources of spiritual knowledge for the Christian, the scriptures are the only infallible source, or the supreme spiritual authority. For example:
Wikipedia: Sola Scriptura (by scripture alone in English) is a theological doctrine held by some Protestant Christian denominations that the Christian scriptures are the sole infallible source of authority for Christian faith and practice.
Marty Foord writing at the Gospel Coalition Australia site: Firstly, sola scriptura meant Scripture was the supreme authority over the church. It did not mean Scripture was the only authority. Luther, Calvin, and the other reformers used other authorities like reason and tradition.
Mark D. Thompson writing at crossway.org: But, critically, both an appeal to the fathers and the application of reason could be questioned on the basis of the plain reading of the text of Scripture. Scripture alone must reign. Our consciences are not captive to any other authority than the Word of God.
Wayne Grudem: In order to guard against making our authority something other than the Bible, major confessions of faith have insisted that the words of God in Scripture are our authority, not some position arrived at after the Bible was finished. This is the Reformation doctrine of sola Scriptura, or “the Bible alone,” as our ultimate authority for doctrine and life.
This doctrine was defined in the Westminster Confession (though without the phrase sola scriptura):
WCF 1.10: The Supreme Judge, by which all controversies of religion are to be determined, and all decrees of councils, opinions of ancient writers, doctrines of men, and private spirits, are to be examined, and in whose sentence we are to rest, can be no other but the Holy Spirit speaking in the Scripture.
Unfortunately there is a real lack of clarify over these definitions, and many other people give contrary definitions. Keith Mathison in his book The Shape of Sola Scriptura says that many evangelical, fundamentalist, or non-denominational Christians say they believe in Sola Scriptura but reject the ancient creeds and any concept of tradition. On the other hand, there are proponents of Prima Scriptura a doctrine which seems at least on the surface to be essentially the same as Sola Scriptura is defined above, but is upheld by its supporters in explicit contrast to Sola Scriptura.
I would like to ask who are the self-professed proponents of Sola Scriptura, who would define it differently to the above definitions, or differently to Prima Scriptura? Who are the people who say they believe in Sola Scriptura, but mean something else by it than what I believe is the general consensus definition above?
To be clear: I am not asking about opponents to Sola Scriptura who might define it differently, either out of ignorance or for polemic reasons, or about proponents of Prima Scriptura.
To make this into an overview question rather than an unbounded list question, it would be good if answers can explain the shape and history of these Christians: do they come from one denomination, or stem from one important teacher or group of Christians in the past? Is there a creed or confession which defines it differently? Is there an important systematic theology textbook or course which defines their doctrine? Is there a theological college which teaches "Solo Scriptura"? In short, is there an organised group of Protestant Christians promoting an alternative definition of Sola Scriptura, or is it just isolated Christians and Churches which may not be aware that they are not teaching either the historical or commonly agreed upon doctrine?