In the interest of cooperation and working together I would rather the terms “developer” and “designer” are not applied this way. The roles in question here are better termed as “programmer” and “graphic designer” (or “visual/aesthetic/UX designer”).
Everybody on a software project is a “developer”, and it’s important to be in that mindset, exactly as you say. Everybody is involved in “design” as well, though somepeople more on internals and others more on visuals.
I recently wrote an article about the role of a programmer, I think it fits well with what you request of them here. https://medium.com/mortoray/im-proud-to-be-a-programmer-800828793c6a