What is it called when someone focuses too much on your rhetoric anecdote?

There are a number of expressions. "pedantic" and "nit-picking" are two of them. You almost used "pettiness." which is good also. I was an editor of my secondary-school newspaper years ago. It took me a few years to overcome the tendency to read as an editor. One purpose of language exchange is to help with details, so be tolerant. Sometimes the corrections are arguable. Cervantes and I wrote/write the name of his masterpiece as Don Quixote, but my friend, a professor in the language department of the University of Buenos Aires, corrected it to Don Quijote to conform to the standard of Real Academia Española.

Written Spanish is similar in the entire Hispanidad, and you may read Latin-American writing by reading major newspapers from the various nation's newspapers. There are wonderful writers of fiction also. Check those who won Nobel Prizes.

Language exchange programs identify the location of the participant. Use Skype to practice different accents. Some of them are quite different even within the same country. The Bogota dialect enjoys high esteem, while that of Medellin is very different. The "ll" sounds more like the English "sh." Medeshin...

Good luck and enjoy,

Pedantic is the most correct word but that's rather formal and, well, pedantic. Many people, me included, call them "English teachers" in that context, which is of course also English language. Not real teachers, just all the time butting in and correcting people's language.