Can hatsu be used with activities, like hatsu-no poem?
Both of them are meaning basically the same.
But in many cases,初 sentence has a subject at the beginning to say whose first.
- 私初のポエム(My first poem)
- Aチーム初の得点(A team's first point)
- 世界初の車(The world's first car)
And the case of 最初,this can also have a subject though,in many cases it doesn't.this does not mean 最初 always pointing "the world's first" nor distinguishing something particular. So You have to read between the lines and guess the meaning.
- 最初のプロテニスプレイヤー(the first professional tennis player)
- 最初の電車を逃したので遅刻した(I was late because I missed the first train)
It's probably good to mention basic difference between them. 初 is a kanji and 最初 is a 熟語(idiomatic phrase) which consists of the kanji 初 and 最 which means 'most'. unlike other 熟語 of 最,最初's 最 basically does not have any meaning but emphasizing the word,Because 初 is meaning 'first' without 最. That's the reason why 初ポエム and 最初のポエム are almost the same meaning.