Fix client project test

[danyalaytekin]

This PR fixes the new client project test by responding to recent evolution of ESLint:

• limits package.devDependencies.eslint to ~8.52.0 because some of this config's rules are considered to be deprecated by 8.53.0 and above. A client project that does not enforce the absence of deprecated rules can drift higher, so package.peerDependencies.eslint has been left unchanged
• changes the new client project test to use eslint@8, because eslint@9 by default looks for a 'Flat Config'. There is a converter which the test could be upgraded to use in future if our config doesn't itself become a flat config.