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I just encountered in the HSK4 book 一份工作 while I'm accustomed to 一家工作. Judging from Baidu (一份工作; 一家工作), both are grammatically correct and widely used.
I am unsure if they're different or not. My guess in this case is that they're the same, but sometimes the measure word changes the meaning.
Question: What's the difference between 一份工作 and 一家工作?
The correct measure word for 工作 is 份, so 一份工作 is correct.
一家工作 you linked in is shorthand for 一家公司工作. In spoken, you might hear one say 我在这一家工作了很久 == 我在这一家公司/单位工作了很久.
一份 is a proper classifier for 工作
一份工作 is grammatical
一家 is a proper classifier for 公司
一家工作 is not grammatical , 一家公司 is
The examples you found about 一家工作 are mostly out of context
[(在同一家)(工作)] = [(work at) (the same place)] - 一家 is a classifier for an establishment that employ people, e.g. 一家公司, 一家工厂 or 一家出版社 which is omitted
[(创立)(一家)(工作时间自由的公司)] = [(establish) (a) (company that has flexible work hours)] - 一家 is a proper classifier for 公司, not 工作
[(我)(上一家)(工作过的公司)] = [(the last) (company I've worked for)] - 一家 is a proper classifier for 公司, not 工作
