Can anyone explain to me the difference between Midrash and Medrash? Is one of them known to be more correct (i.e. most likely older)?
To the best of my knowledge, it's just different way of saying the same thing: "expounding" on a verse. The Hebrew word would always be vowelized "midrash." But at some point in history (most likely involving Yiddish), the word has often come to be pronounced "medrish" or "medrash." We all speak of a "Beis Medrish" or "Medrish Rabba Chapter 5", I'm not sure how that occurred colloquially, but technically the Hebrew word is always "midrash." Your choice of how to pronounce it will likely depend on context, what language you're speaking, and whether your audience will appreciate the colloquialism or proper form.
Technically, midrash is Hebrew and medrash is Yiddish. Hebrew is older.