you're actually both kind of wrong.
free verse is a form of structure, nothing to do with free writing. it forgoes classical ideas of structured verse, for example a sonnet having 14 lines of strict iambic pentameter, in favour of a free wheeling approach to lines/verses/general form. it became popular during the early twentieth century, championed by t s eliot, the beat poets and adrienne rich.
free writing is often a means to an end; a method of renewing interest and ideas, though sometimes used for therapeutic purposes. it entails forgoing conscious thought, allowing the pen/your hand to write, without any thought as to what you are writing. the moment you break from this, and start correcting/applying structure or modifying it in any way, you are no longer free-writing.
