I don't think you're listening to the response.

What someone wants from a job offer is for the company show that they genuinely want them to work there. That they are willing to give the candidate a bit more, because they are THE right person for the job. Ideally, they want to be blown away by your generosity. The offer should exceed their expectations.

The response you provide here feels like quibbling. It feels like 'we want you here, but..' Yes, the person wants more and you are right that it's not really about compensation. It's about feeling wanted.

The operative word is not 'fair' it's 'feels'.

The company's opening offer has not made them feel like you actually want them, and by nit-picking over their definition of 'fair' you're about to lose them completely.