I checked the definitions in here for 生産 and 生出 and there appears to be a huge overlap between the two words. That said, what is the major difference between these two?
産出【さんしゅつ】: production (of natural resources and agricultural crops)
生産【せいさん】: production (mainly for industrial products, but also for natural resources and crops)
(Strictly speaking, as defined in the dictionary you checked, 生産 can be used for human reproduction in highly academic contexts. For example 総再生産率 means gross reproduction rate. But not many native speakers are familiar with this usage, and it's wise to always avoid using 生産 for humans unless you are talking with experts.)
(出産 is also "birth" but it's closer to medical "delivery" as the last process of pregnancy.)
輩出【はいしゅつ】: production (of capable people from school)
生出【せいしゅつ】 is an old word and it's extremely rare in modern Japanese. You probably don't have to remember this. (My IME and I did not know this.)
生出 might be a psuedo-Sino-Japanese compound that derives from 生み出す. It'd sound weird in the Chinese sense of the character 生. May 5, 2020 at 4:38