I obviously am familiar with the texts mentioning it is an average lower bound etc... but still wondering why the word amortized was put there ?
Why is amortize used in describing algorithm analysis ?
Because the computer scientists who thought up the idea were using a financial analogy.
You amortise a significant expenditure (like building a new house) by paying for it over time (perhaps with a mortgage, which has the same root).
Similarly, in amortised analysis of algorithms you pay for a huge and uncommon occurrence (copying an entire vector of when it gets full) by spreading its cost over subsequent operations (or previous operations in the banker's model).