Although as the GENUKI page points out, Chester was a County Borough from 1889, and thus in an administrative sense became separate from Cheshire. (Even though the County Hall for Cheshire remained in Chester!)
Something similar can be seen in parish registers when entries may be headed by (e.g.) "County of Bristol" instead of "County of Gloucester". However, since the official "County of" form of Cheshire's name is "County of Chester", there wouldn't be any visible difference in Chester's case.
In ceremonial terms, it would still be regarded as being in Cheshire. (If you want to know what that means, ask who would greet Queen Victoria on her visits? - answer, the Lord Lieutenant of Cheshire, no doubt with the Lord Mayor breathing down his neck).
The relic of the administrative separation seems to linger even today with the unitary authority's name of "Cheshire West and Chester". I do tend to think, "Where else would Chester be, other than in Cheshire West?" but presumably someone wanted to emphasise that Chester was a bit different!
