> I always found this statement to be rather wishful. Individual lowering of prices makes sense if and only if your competitor is capable of saturating the market. Otherwise, demand elasticity becomes very relevant. Sure, your competitor may take the larger share of the market, but then you can compensate with higher per item profit.

You should check the distinction between Bertrand and Cournot competition. Bertrand competition is price competition where the competitor can saturate the market, as you mention. Cournot competition, on the other hand, captures your intuition of competition on quantities rather than prices.