In general, "verbetering" does mean the same as "correctie". Both words mean "correction". Although, in some cases, "verbetering" can mean "improvement", in the way that what is improved, or "verbeterd", was not wrong, it just could be better if altered a bit. In the case of "bedankt voor de verbetering" the words "verbetering" and "correctie" have the same meaning, as the person who uses the centence probably made a mistake and was corrected by someone. As for the word "verbetering" being a 'more Dutch' word, I would agree with you, as far as the origin of the word is concerned, since the word "correctie" comes from the French word "corriger".