This is my guess:
amortized because we want to know the "averaged" complexity over
n operations assuming the operation is going to be used frequently
unamortized when you know the operation is going to be used rarely
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You can find the motivation for the amortized analysis from this reference:
The motivation for amortized analysis is that looking at the worst-case time per operation can be too pessimistic if the only way to produce an expensive operation is to "set it up" with a large number of cheap operations beforehand.
Hence, unamortized (asymptotic) analysis means considering time complexity of an algorithm (instead of each operation) when each operation can be counted in constant time.