To be a pedantic here, I don't think term amortized average cost is useful.

For what I've understood, average cost means average over all possible random inputs. So for specific set inputs, the cost can be larger than that.

Amortized cost means the average cost of the operation over any combination of inputs. So for specific set of inputs, the cost is the stated.

For hash table insertion, both the amortized and average cost can be O(1) with linked-list confliction resolution, but whereas average time of a lookup operation is O(1), amortized is larger (I don't know if there's analysis on this, but O(n) seems correct by gut-feeling).