Congrats on your upcoming trip to Italy, Bucephale!
I heartily endorse fellow posters, and I encourage you to get the Eat Italy app. Well worth the investment to buy Elizabeth Minchilli's Rome, Florence, and Venice city guides. Elizabeth has impeccable taste, and her recommendations really are the best inside tips of eating around the city. I loved discovering the Roscioli bakery on the Esquiline hill. Delicious food and great memories now that Roscioli's bakery is no longer an option for me (subsequent celiac diagnoses change things!).
Another fantastic app that I highly recommend (and that is free!) is Natalie Kennedy's Discover Rome app. Natalie writes a delightful blog, and she constantly updates her food guides - both on her blog and in her app. You are sure to get great recommendations that are NOT tourist traps. She even wrote an entire blog post on eating well - and avoiding tourist traps - around the Colosseum.
Great find on the Eater map! As Suki pointed out, Katie Parla used to have a Rome app. Unfortunately the app has not been updated to accommodate current smartphone technology, and as such is no longer available. In the interim, DEFINITELY comb through Katie's Eater map that you referenced in your original post. Great recommendations. Eater recently published a large guide to Rome, covering many food topics. Well worth checking out.
If you are looking for fun food ideas in Florence, consider a food tour with Curious Appetite Travel. I did their Medieval and Renaissance Food & Spirits tour back in February. One of the highlights of my trip! Loved every minute, it was so different, and I learned so much on a very unique topic!
Let us know if we can help further and happy planning!