The name tells you what the room was for. Beit is house. Midrash is inquiry. Put them together and you do not get a lecture hall, you get a house of searching, and the difference is the whole argument.

The phrase does not appear in the Tanakh. Its earliest attestation is Ben Sira, around the second century before the common era, where the teacher invites the untaught to lodge in his house of instruction. The practice it describes is older than the phrase: Deuteronomy 6:7 has Torah spoken when you sit, when you walk, when you lie down, when you rise, which is a description of continuous communal conversation rather than a scheduled monologue.

Two things kept the room honest. Everyone brought something, and everything brought got weighed. Proverbs 27:17 names the cost of that, iron sharpening iron, because sharpening requires friction and friction requires being contradicted. A room where nobody is ever contradicted is not a beit midrash. It is a place a man goes to be agreed with.