There are a few extremely interesting questions asked here, including "what's the origin," "who popularized its usage," and "when did its meaning change from major triad to major 7." I'll try to shed some light on the question in the title, "what's the origin?"
If you draw an equilateral triangle inside the circle of fifths, its vertices connect an augmented triad. These and other similar triangle-shaped drawings within the circle of fifths were well known long before the 1950s, and they are documented within classical music and music theory. I haven't seen any evidence that these drawings were or weren't known by the relevant jazz musicians in the 1950s, but I haven't done much digging. I do know that Charlie Parker discusses studying classical theory in his interviews from the 1950s, which suggests he and perhaps also other jazz musicians could have been familiar with pictures like the one below.
The first documented usage that I have seen of the ∆ symbol in jazz is from Charles Mingus in 1957. The Civil Rights Digital Library contains this entry for a trumpet score handwritten by Charles Mingus and dated 1957 according to the record:
In the B section, we see these chords:
A♭–∆7. As you surely know, the minor major seventh chords contain an augmented triad like the one drawn inside the circle of fifths. In
C–∆7 the augmented triad is
E♭ G B. Perhaps the equilateral triangle symbol, which earlier in history was synonymous with an augmented fifth, was first used in jazz to indicate the major minor seventh chord (e.g.,
C–∆7)--a chord which at the time had no other symbol associated with it. This is consistent with the documented history of the ∆ symbol, which shows its first usage in a minor major seventh chord, not a
In one of the links you included, Matt L. cites the book The Chord Scale Theory & Jazz Harmony by B. Nettles and R. Graf as evidence that the ∆ symbol originally meant "major triad." But that's not what the text states:
the ∆ originally meant "triad" and if written with haste may appear as a circle. (Nettles and Graf, p. 23)
This is in a section about common chord notations. It's not a history book, and Nettles and Graf do not provide a citation for their claim. They also don't clarify whether the original meaning of ∆ was major triad or simply triad (as in augmented triad). So I don't think that text is very useful in answering your question about the history and origin of the ∆ symbol in jazz.
Just to clarify, I'm piecing together parts of history to try and provide a possible explanation. This normally wouldn't be appropriate, but I think it's useful in this case since it would appear that the only way to answer the question about the origin of the ∆ symbol in jazz is through some original research. Interesting areas of further study would include some of your other questions, along with some other question like "was Mingus/Coltrane familiar with drawing the triangle inside the circle of fifths?" To uncover more of the history, perhaps it would be instructive to look at historical records of songs from the 1950s that contain major minor seventh chords.