Does qin1 cai4 translate to bok choy?
You probably mean qing1 cai4 (青菜). It is used for green vegetables in general, but is also used for Chinese cabbage.

现代汉语规范词典 has this to say:
1.
名
一般指小白菜或油菜。
2.
名
泛指蔬菜。

小白菜 is typically translated as Chinese cabbage or bak choy in dictionaries and also used like this in  many places in China, in other places people often mean baby bak choy when they use 小白菜.

https://en.wikipedia.org/wiki/Chinese_cabbage#Commercial_variants_of_Chinensis has this to say:

Baby Pak choi, Shanghai Pak choi, or mei quin choi (Chinese: 上海白菜; pinyin: Shànghǎi báicài; Japanese: 青梗菜, chingensai) refers to greener varieties where the varioles are also green. It is simply a less-mature version that could develop into the white-stemmed variety with more time to grow before being harvested. In Shanghai and other eastern China provinces, it is simply called qīngcài (青菜; literally blue/green vegetable) or qīngjiāngcài (青江菜; literally "blue/green river vegetable").