In most of these lines 자 is the 'let's ~' suggestion sitting inside a bigger sentence rather than ending it: 디트로이트라고 부르자 했죠 'we said, let's just call it the Detroit', 회식을 없애자 이 말입니다 'what I'm saying is, let's scrap the company dinner'. One line uses a genuinely different 자, the one meaning 'the instant X happened': 통화 끝나자 딱 마주친 세찬 'Se Chan turned up the second the call ended'. The examples mix the two uses, so what unites them cannot be pinned down from this evidence alone; the safest reading is to check whether a 'let's' makes sense, and fall back on 'as soon as' if it doesn't.
Register: casual
The examples below did not fully pin this one down — read them before relying on the entry above.