Annie. The Medium app on my iPad is identical to the Medium app on my iPhone (just taller and wider due to the larger screen.) It is the newest version. I update my apps almost every day.
On the bottom of my Medium app (on both devices) I have a row of five icons. A Home (house) icon, a Bookmark icon, an Editor icon (looks like a sheet of paper with a + sign), a Notifications icon (looks like a bell), and a User icon (person’s head and shoulders). The center one is the Editor icon. I allows you do what I described earlier.
I just went through the process of creating a test story again (I don’t normally use it) and published it. I then quickly deleted it since it wasn’t really a story.
I discovered that the “add to publications” button only allows me to add stories to my own publications (One Minute Wit, Two Minute Tales, Starlin Serials), so no, the app won’t allow you to post to someone else’s publication. You will have to use a computer to submit to a third party publication.
But you can publish to your own profile page from either an iPhone or iPad. I just did it.