The challenge with dictionary definitions is that they don't always give a feel for appropriate nuance or context. Sometimes, though, the word itself can give you useful information.
Melodrama is a good example, as it contains the word "drama".
Drama: a piece of writing that tells a story and is performed on a stage, a play, movie, television show, or radio show that is about a serious subject and is not meant to make the audience laugh (as compared to "comedy")
So, (among other definitions) a drama is an artificial experience, like a movie or a TV show, written to create an emotional effect. Something that is melodramatic does this in an exaggerated or overly stylized way. The director may have meant for the exaggeration to increase the emotional effect, but actually it just makes the drama look silly and even more artificial.
In the situation you describe there is no drama -- no movie or TV show, no emotional setup -- so you wouldn't ordinarily call it melodramatic.
Now, it is possible for someone to act melodramatic (overly emotional) but this implies that they are acting as if they are in a bad movie. They are exaggerating their actions for emotional effect.
So, again, in the situation you describe: while there is an exaggerated claim, there is no drama.
Examples of shows that are frequently melodramatic: movies in which one of the characters is sick or dying, movies about significant real-world tragic events, and daytime "soap operas".