You can use a regular microfiber for cleaning the interiors without any issues. I have a large assortment of microfibers on hand and have 4 or 5 different colors. I use a different color for different things; wax, polish, interior, carpets, tires. Any microfiber will do but you just dont want to grab a dirty towel used on the exterior and then use that for the interior. Last week I went to clean some leather seats and I accidently grabbed a towel I had just used on the exterior. My client called me back a couple hours later to complain that her seats felt sticky and still were filthy.
The white towels you mentioned do lint a lot so I only use microfiber but this too also creates fuzz balls at times depending on the brand and how it was washed last. The best towels to use for interiors are the suede microfibers as they don't lint or shed. Carpro has some great ones available here on autogeek that I thoroughly love; theyre also great for cleaning windows with too!