If the engine turns over and catches, but won't run smoothly or stay running, the battery isn't the problem. (If it were the battery, it wouldn't turn over at all, or it would turn over slowly.)
To EOC with the key, you briefly switch from the RUN position back one click to the ACC position, and after the engine quits, switch forward again to RUN.
But I'm going to say again I wouldn't do it with this car until the re-starting issue is sorted out. It could need some basic tune-up items - plugs, cap, rotor, etc - before it will restart reliably.
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