How to do if a potential function Does not work? Amortized analysis
When you delete, your $actual$ $cost$, or $c_n$ by your notation, is $log(n)$.

it means you want: $\Delta \phi = \phi_{n+1}-\phi_n \approx -log(n)$

Its not hard to find such potential function that satisfies both:

try $\phi = nlog(n)$

for insert:

$Cost_{insert} = log(n) + (n+1)log(n+1) - nlog(n)$

$= log(n)+log(n+1)+nlog(n+1)-nlog(n) = O(log(n))$

$Cost_{delete} = log(n) + (n-1)log(n-1) - nlog(n) $

$= log(n)-log(n-1)+nlog(n-1)-nlog(n) $

$\leq (log(n)-log(n)) +(nlog(n)-nlog(n)) = O(1)$

All potential functions that satisfy $\phi(i) \geq 0 $, $\phi(0)=0 $ are correct, they just may not prove the amortized time you have in mind.