This is an interesting theory you've put together, but unfortunately there are some flaws. The biggest issue is that the this is not a continuum of an evolving necktie, but different types of neckwear - the early seventeenth-century collar isn't a proto-necktie.

In the day of Martin Luther - say 1517-1546, his life post-Theses - Northern European men simply didn't have separate pieces of neckwear. Men wore shirts of linen that typically came up to the neck, sometimes with a plain hem and sometimes with a standing collar, and sometimes with a ruffle. There is a general correlation of rich men with ruffled collars (as fine white linen was a sign of status), but it's not a rule, and there's no indication that the decision to have a visible collar related to anything philosophical. Martin Luther doesn't have a necktie because nobody in the period had a necktie, and his lack of white linen at his neck likely has to do with his social position and/or the fact that his doublet has a tall collar that would obscure one.

In the late sixteenth century, the ruffled collar became the ruff, and then by the end of the century, the flat falling collar which you associate with Puritans. In reality, it was simply a fashionable, then an ordinary piece of neckwear that was worn by both political wings. I'm not aware of anyone wearing them different lengths on different sides - where have you gotten this?

It's at the end of the seventeenth century that we see the introduction of what can be considered the proto-necktie. I'm going to copy/paste from a previous answer of mine on the subject:

The descriptions you've seen of the cravat being taken from Croatian soldiers in the 17th century are commonly accepted by dress historians. It continued to be a rectangle or folded square, with or without a significant portion of the ends being lace, tied around the neck for several decades, but in the early 18th century men began to wear a "stock": a stiffened band buckled around the neck, covered with very fine linen. This could be worn with a smaller cravat tied over it, but more frequently men began to instead have a ruffle sewn down the front of the shirt instead, to poke out from the openings of the waistcoat and coat - but by the end of the century, a tied cravat began to come back into favor, worn over only the turned-up collar of the shirt, rather than a stiffened stock. (In the British military, the stock was black, and once the stock was gone a black cravat was worn over the collar.)

This led into the situation you may have seen satirized in the early 19th century - the shirt collar itself grew and was starched in order to stand up straight behind the cravat. In the 1820s and 1830s colored neckties became more popular (as did the word "necktie"), and by the 1850s were most commonly worn in ordinary dress, with white neckwear relegated to formal dress. The colored tie was worn with what the upper and middle classes considered everyday businesswear or casual dress. By the end of 19th century, the method of tie-tying most commonly in use today had been devised.

I'd suspect that Jackson, in the portrait you've linked, is simply wearing a black cravat that cannot be seen against his black coat collar and black waistcoat. It might look quite a lot like Clay's, but it's hard to compare clothing realism between a photograph and a stylized, romantic painting.

The reason everyone in the 1860 montage is wearing the same style of collar and necktie is that that simply was the stylish way to wear them. Grover Cleveland likewise is shown sporting two different styles of tie-tying with no connotations. Religious and political stances have only ever been associated with neckties in subtle ways - flamboyantly colored ties have been associated with counterculture in various periods where men normally wear staid colors, for instance.