Latin america is iffy because its referring to the latin language based areas of the americas which is a result of colonialism. The only term I actually have a problem with as a concept is old world.Central "America", "New World" all have far more colonial connotations than something like "Latin America." Being part of Latin America is a point of pride for the people living there as their primary square focus is US imperialism, being apart from the US of America. Spanish colonialism is far in the rear mirror, not a part of living history, and the people are pretty much assimilated into a hybrid culture.
I have never heard of anyone from Latin America objecting to its name usage, even among still-existing tribes.
Also what kind of mental gymnastics did you do to determine central america has a more colonial origin than the other 2. It is literally just referencing that the region is at the centre point of the americas which is a defined continental region its no different than saying south east asia and has a much stronger biogeographical connection than latin america.