Something was off.
For six months, they had held onto something which they had misconstrued the intent.There are always three sides to a story, yours, theirs and the truth.
Problem is, unless you record everything, it is difficult to defend either position which is why when you say something, particularly if it is sensitive, ensure you speak with clarity and context.
What exactly do you mean, and what is the context in which it is important.
Here is a great place to use stories by way of example, to further iterate what it is you are trying to convey.
If need be, ask them to use their own words to replay what you had just said.
It may just save some heat later on.