It's really context driven. You don't declare you're "home" when you're entering someone else's house. You could, but it's a little off. Instead, you might say something like ojamamashimasu or hairune. If you've been on a trip and upon returning, meet your friend at the station, you might say tataima, in the sense that you returned to your home city, but it's not first or the most common thing you'd expect to hear. Even less at your friend's house.
In that scene, the correct translation is more than "I'm back" and maybe a little less than "I'm home." But between those two, it's more "I'm home" than "I'm back." You can also tell that it slightly confuses Doubara. It speaks to Mio's state of mind as far as where she thinks she belongs, and where she thinks she can find support. So it's a very loaded, extremely significant dialogue.
