re: "...I put a multi-meter on the battery and (after scrapping some gunk off of the terminal tops) got a reading of 12V and with the engine running I get about 12.9V. I have also checked and topped up the distilled water levels in the battery...."
On the face of it, this (not being able to restart the engine after even a brief shutdown) is typical of a battery dying from old age -- if your battery is original (your car is a 1999, right?), it's lived a long and fruitfull life -- and has died at a reasonable (if not better than) lifespan.
But let's consider that it's newer than 1999...
If your battery was fully charged, it would register about ~13 volts just after you shut down the engine, and about 12.6 volts (give or take a bit) a few hours later. I don't know if you were very roughly approximating your voltage when you wrote "12V", or you were being exact -- if exact, that battery is very run down, as 12.06 volts represents being only 25% charged (i.e., 75% empty)! (BTW, the best way to measure a battery's charge is with a hydrometer, but it seems you don't have one.)
On the other hand, most batteries, even bad ones, should have much more than 12.9 v when the engine is running. The right voltage when firt starting up in the morning (with everything cold) would be in the high 14's -- i.e., 14.6 v perhaps -- even higher (~15 v) in very cold mornings (the max. charge is inversely temperature-dependent). As the engine (alternator) and battery warm, it's OK to drop to low 14's, or even (in very, very hot weather) as low as high 13's (e.g., 13.8 v).
It is possible, though, that your battery is so bad that it can't even accept such a high charge (maybe it's developed an internal short), and that's why your alternator can't get it higher than 12.9 v when it's running.
What I'd do, unless the battery is very new, is to install a new battery, expecting that the problem was just old age. If then the voltage (with the engine running) climbs to around mid-14 v or more, then your charging system has probably been good all along and your battery was just dying of old age. Everything should be good from then on.
If your current battery was fairly new, however, then it died prematurely, so take the car to a reputable shop to check the charging system, because it shouldn't have gone bad unless the charging system wasn't working right.
Good luck.
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