I sympathise! I think we are advised to avoid "Ultra-processed" food rather than just "processed" food since more or less everything is "processed" before we eat it. Chopping vegetables is processing them.
I believe food is now classified as whole/natural/whatever (fresh fruit & veg etc); processed (butter, cheese, etc and anything natural just cooked or canned) and ultra-processed which is mostly stuff which has a long list of ingredients which most people don't understand and wouldn't keep in their family kitchen cupboard. The difficulty is in deciding where the boundaries are and whether it's important.
Philadelphia - and I think the Tregar and Finlandia cheeses too - contains milk, salt, guam gum and citric acid. I don't have guam gum in my kitchen but I often use a dash of citric acid in my bread recipes because it helps the bread rise. And talking of bread: a true sourdough loaf just has flour, salt, water and the massa madre starter but have you ever looked at the list of ingredients on an ordinary factory-made. shop-bought loaf of bread? Now that's what I call ultra-processed!
Incidentally, to drag this back on topic, if one has the time and the inclination it's not that hard to make one's own Philadelphia style cheese in the kitchen and forget about the guam gum and the citric acid. I regularly make my own ricotta - which is even easier.