Normally, when discussing amortisation and worst-case complexity, amortisation negates the worst-case scenarios, and the BigO describes the average for the operation (the way it's used in interviews nowadays).
For example, when an insertion of an element requires reallocation and copying of the array, increasing the size in two, it is still said that insertion is $O(1)$, not $O(n)$; the worst-case performance is rarely mentioned. Compare this with sorting algorithms, where the worst case could be $O(n^2)$, whereas the average case could be $O(n\log{}n)$, and this is always specified and discussed.
What is the terminology to express this N-level complexity for the $O(1)$ insertion that requires reallocation with copy of $O(N)$ elements?
What is the alternative terminology for expressing an optimisation approach, with delayed copy, where the worst-case complexity remains constant-time, provided the operation to allocate new chunk of memory itself is constant. In other words, where you keep both arrays, and do a lazy-copy approach.
Specifically, both of the above could be described as amortisation. But I'm having trouble finding out the absolutely correct terminology for this nuanced problem, as how do you distinguish between these two different scenarios in describing amortisation?!