Terry, you need to stop doubting yourself, you're a hell of a lot better with mechanical diagnosis than you used to be! We're all here to bounce idea's around, and yours are as good as any nowadays

Anyways, in regards to the problem, I had the same issue with my Si when I first got it. Would start up fine from cold, but once I'd driven for a while and got up to temperature, turn it off, then it would struggle its tits off trying to start!
Took the alternator off and got that checked, all fine. Wires were fine. Starter was fine. Turned out it was indeed the battery! If you can get a nice fresh one to borrow, try that first before you go dismantling things

If that doesn't cure the problem, then I'd be looking at the alternator and starter motor

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