What wording would be correct to say “I can reduce the complexity from O(n^2) to O(n)” but reduce in algorithm analysis means you can cast one problem in terms of another for which there exist a known solution. Therefore, I think is more correct to say “I can improve the complexity from O(n^2) to O(n)”. Right?
Separate from this if I improve the complexity from, e.g., O(n^2) to O(n) would this be a correct way to say it? I improved the complexity by “one order of complexity”? or by “one or several degrees of complexity”?
In answer to the question in your first paragraph either reduce or improve is acceptable. If you hold fast to the idea that reduce necessarily implies recasting a problem then stick to improve. Personally I don’t see that reduce does necessarily imply recasting a problem, but I’m often wrong in these matters.
As to your the question in your second paragraph, I don’t think that the phrase one order of complexity is well-defined and you should, therefore, avoid it.