I don't see anything suggesting the battery was bad.
However the volt meter makes testing a battery so simple.
Find a problem before trying to fix it - which is why that
voltmeter is a so important tool.

Run car even at idle without electrical loads. Battery
voltage should be above 13 volts. Turn car off. Battery
voltage should be somewhere in the neighborhood of 12.5 to 13
volts. Turn on high beams. Battery voltage should maintain
over 12 volts and over many minutes, never drop below 12
volts. That is a good battery.

Described previously is what I would expect from an
alternator problem. Alternators really are three separate
electrical generators. If one fails, then output voltage will
appear low during low RPMs. This is why smarter mechanics have
an oscilloscope). Alternator will charge battery under most
conditions but will slowly discharge battery during a full
load of electrical peripherals (headlights, read window
defogger, heater fan).

That idiot light can be a problem. An alternator that
outputs power will still discharge battery. Idiot light
properly reports that alternator is outputting electricity
(even if that is not enough electricity). Same problem can
also occur if fan belt to alternator is slipping. Great
profit to the local repair shop. They sell a new battery
since - as demonstrated here - knowledge of how alternators
can fail is so little understood.

A car battery that does not last at least seven years (never
even garaged) suggests a car problem - either in design or a
maintenance problem. Demonstrated are two reasons why an
alternator can suggest battery failure - slipping fan belt
that makes no indication OR loss of 1/3rd of the alternator
(usually a diode failure inside the alternator).

Tibur Waltson wrote:
> Attn: Everyone,
>
> The 3-year old low-price, drained-several-times, weak-battery
> described above is now replaced with a new 7-years warrantee battery.
> Let's see how long this one will go. I have another similar
> alternator question for a second car, if I may.
>
> Recently my wife's '97 BMW 318ti's alternator was replaced and it
> shows 13.7V at any rpm with no loads. With loads it's 13.6V at 2000
> rpm or 9-10V at idle. Engine off it reads 12.6V. Alternator is Valeo,
> made in France. Battery is two years-old. Is the alternator fine?