"I write down the recipe."
Translation:אני רושמת את המתכון.
not only it sould be accepred, it's far mor correctly than "רושם". The verb רושם refers more to the "picturesque" aspect of the writing / drawing (such as face drawing, calligraphy, etc.) wile כותב refers to letters, so as long as one actually write (and does not type or something like it) רושם isn't fully wrong, but at the same time it's not exactly accurate
for me, I learned as anyone herr that רושם has to do with writing something down, like noting, booking something, whereas כותב is just the act of writing in general, that's how it works in my head, if it wrong, let me know. I stand to the usual, don't want to sound perfect speaking hebrew, but natural.