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It's not the difference between an n-bit random number and an n-bit modulus. It's the difference between a 512-bit random number and a 521-bit random number. It's very simple, but wording it as number vs. modulus is needlessly confusing, just adding to the problem you are bemoaning.

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The issue with cryptography is that you have to be precise, that means the communication needs to involve far more detail, even if it can initially be confusing.

This is one of the major reasons that crypto is hard and if you try to get around the "hard" bit your "shortcut" will probably come back to bite you. When it comes to crypto and accuracy (and hence security), more communication, and detailed communication are probably the solution not the problem.

Agreed; my problem was that the parent comment was imprecise, and hence confusing. "n bit random number vs. n bit modulus" implied n was the same in each case, whereas 521 is not the same as 512.

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