time linearity of LTI system
Your notes gave a counter example, you just proved it in the general case.
Your note used your x[n] as delta and showed that
y[n-1] unequal to H{x[n-1]} which is exactly what you showed.
In the notes example H{x}=0 for all n, as discrit delta equals zero foll all n but n=0, which means H is zero every where for all input of the kind delta[n-n0]  (becouae of the n factor which zeros y at its only non zero point)
Than you have zero output for delta input. But as you showed time varied output isn't zeros every where.