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That sounds exactly right to me as far as the maths is concerned. (I am however not well informed about the philosophical history of the terminology.)
We discussed a similar point at the Cafe. I think an important source of the terminology is the nineteenth century distinction between synthetic geometry (done with figures) and analytic geometry (done with coordinates) (see, e.g., here). Someone like Sophus Lie is using geometric intuition to do his work, and this includes intuition of infinitesimals.
For Kant, all of mathematics concerns the synthetic a priori, arithmetic relies on the inner sensual intuition and geometry the outer.
SDG is not elementary (inside a topos). The Kock–Lawvere axiom involves an exponential object, which is not first-order.
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