The problem is that we all know that spiders and insects are quite different invertebrates, so the choice of 'proper' itself is slightly odd. Otherwise, that adjective (when it has that meaning) can fit either side of the noun.
(Hint: if you try to bump your own posts, it shows only as a thread with one response, so it appears answered to many of us; hence, less chance of getting a real
If that is British English, the American version would be, "The spider is not a real insect." That is a true statement. If that is American, it means that the spider lacks characteristics that more representative insects have, and that is not a true statement, t