I'm a little confused here: in contemporary Hebrew the difference is simple - a box that opens on the side [and has a door] is called a cabinet (ארון) while a box that opens on the top is called a box (תיבה).
Even-Shushan dictionary holds a similar opinion. It is pretty much consistent with the Mishnaic use of those words (Masechet Shabbos, Kelim and more), but in the Torah, it is hard to tell, for example, Aron Habrit that was a box or Yosef's coffin is also called Aron vs Tevat Noah or Moses' Tevah.
Do the interpreters address this difference?