This picture illustrates the difference between the different dragon and dragon-adjacent creatures! My partner is also pretty adamant about this subject, so I've wound up developing the same pedantry.

They are synonymous. Throughout most of history, the word "dragon" hasn't been used to refer to a specific form of big reptile. Medieval sources aren't specific, and artwork is all over the place on what they look like: how many legs, how many wings (if any), etc. It's only through the relatively recent phenomenon of trying to classify and systematize things that people have adopted this rigid view of the word. My guess is that a big part of it is games like Dungeons & Dragons becoming more popular.