This 才 dates verb A to a moment barely past (only just stepped out); 就 dates verb B to right after, the consequence already arriving. They mark different verbs in the sentence, not rival readings of one.
The other 才 says the step came later than expected, still slow to arrive (等到八點他才來). This 才 says the step came only a moment ago, fresh and barely done. Success: 才 stamps verb A as just-now. Failure: 才 stamps verb A as overdue — wrong point.
他出門才,就開始下雨了。 → 他才出門,就開始下雨了。 (才 sits before the verb 出門, not after it)
他才出門了,就開始下雨了。 → 他才出門,就開始下雨了。 (no 了 on the barely-done verb A under 才)
他剛才出門很久,就開始下雨。 → 他才出門,就開始下雨了。 (才 marks the moment as fresh; a long gap contradicts it)
English 'just' floats and can stack ('he just only just left'); Chinese 才 is fixed before the verb and carries the just-now sense alone, no extra adverb needed.