The latter. That's always what "flattening the curve" has meant. The total area under the curve (i.e. total unique infected individuals) stays about the same.
"Flattening the curve" is not the only possible outcome.
A cure, an inoculation, or segmenting geographic regions until localized herd immunity (like New Zealand just did) are all examples of cutting the curve short.
Vaccines are far away from approval for widespread use, and New Zealand's situation as an island nation is not applicable to something like 98% of the World's population and quite frankly, irrelevant to the discussion.
