So you had a day's running around with the new battery BEFORE the charge light came on? In that case it's probably unrelated to the battery replacement.
Batteries can die suddenly, it's no mystery - they just get old and the plates fall apart. Alternator brushes also wear out, connectors work loose, electronic components fail, etc. Forget about the new battery and just diagnose your current fault.
Are you sure it's the "battery discharge" light that's on? I seem to remember that when the alternator light comes on it also turns on a couple of others, but I might also be thinking of a different car! Some MR2s also have a battery electrolyte level light, could it be that coming on?