Pointcount — the series
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One, two and three point are one construction
The names count how many vanishing points sit at a finite place in the picture, and the count is a fact about how the object is turned and the camera is aimed. Nothing about the method changes between them, and a vanishing point does not appear — it arrives from infinity.
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The arrangement the count cannot see
Five posts laid out five different ways give the same leverage to a sixth and the same separation limit to a quarter, and the sixth arrangement — posts strung out along their own shadows, built to be exactly degenerate — is no worse than the rest. The degeneracy belongs to the family the count does not use, and its conditioning there is exactly zero.
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The third point put where it looks right
Three-point perspective is taught as two vanishing points plus a third placed by judgement. The third is not free: two points and the centre of the picture fix it exactly. It survives being placed by eye because in the layout a book draws it belongs thousands of pixels off the paper, where 400 px of error costs less than a degree — and in a photograph taken looking up at a tower the same 400 px costs ten.