These four get used interchangeably in briefs, in pitches and in most of the articles written about them, which is one reason so much brand work starts out confused and stays that way. They are not synonyms. They answer different questions and they are produced at different points.
Positioning answers: where do we stand, relative to others. It is comparative by nature. You cannot have a position in isolation, only a position relative to a category, a set of alternatives and a buyer. Positioning is mostly an internal strategic decision and it is rarely expressed to customers in its raw form.
Proposition answers: what is the single idea we offer. It is the sharpest external expression of the position, reduced to one thing. Where positioning is a choice about territory, proposition is a choice about language. A proposition can be said. A position can only be occupied.
Purpose answers: why does this organisation exist beyond profit. It is about motive, not market. Crucially it makes no reference to competitors, which is what makes it so easy to write and so often useless commercially. Purpose can support positioning. It cannot substitute for it, and when a business substitutes one for the other you get the familiar result: a beautifully written reason for existing that gives a buyer no reason to choose.
Promise answers: what can a customer count on, every time. It is about consistency and it is mostly operational. A promise is a standard, and the useful thing about standards is that they can be broken, which means they impose obligations on the business rather than just on the marketing.
The relationship, in order: position is the decision, proposition is the expression, promise is the obligation, purpose is the motive.
Where it goes wrong in practice is nearly always the same substitution. A company is asked for its positioning and produces a purpose, because purpose is more comfortable. Purpose asks you to describe your motives, which are flattering and uncontested. Positioning asks you to choose a place and exclude people, which is neither. Given the option, most organisations will write about why they exist all day rather than decide who they are not for.
You can spot the substitution quickly. If the document contains no reference to a competitor, a category or an alternative, it is not positioning, whatever the cover says.
A second, quieter error: treating proposition as a tagline. A proposition is the idea. A tagline is one possible articulation of it, usually written later, often by someone else, and frequently not needed at all. Plenty of strongly positioned businesses have no tagline. Very few have no proposition.