In the CRX world (those only go up to 1991, but should be similar to our 93 in many ways) the main relay is a very frequent suspect for any starting problems. After a couple of decades, the solder joints inside the relay can crack and get intermittent. It is often recommended for owners with problems like yours to take the main relay out, remove the plastic cover, and re-flow the solder or even completely re-solder the connections.
I wound up buying a new relay, myself. In my case, it didn't solve the very intermittent problem, but it has worked for a number of people.
-soD
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