Actually, you are all correct - partially. When flaps are lowered, the wing angle of attack is increased along the span of the flaps. The wingtip angle of attack becomes less than the root, the root stalls first. Increased lift by flaps only goes so far, then lift no longer increases and induced drag increases. On a Cessna 172 for example, you get more lift than drag up to 20 degrees of flaps. 30-40 degrees gets you loads of drag and you can drop into a short field easily. If you have to cob on the power to go around, you have to dump the flaps only to 20 degrees, then milk them up the rest of the way slowly.

Long story short - flaps increase the AOA (and the lift), causing the root of the wing to stall before the tips.