They're all the same of meaning. You can choose any one of them if you like. But 第一场雪 is more common in conversation. 初雪 mostly used in article or speech.

第一场雪=初雪, 第一场=初=first。 in speech or in writing都可以，放心使用不用担心。

Yes, 下初雪 can be used as well, and believe me, people from Mainland China use this phrase often. However, I often use 第一场雪 after the verb 下 personally, like I mentioned days ago. It doesn't mean that you can't say 下初雪. Sometimes learning Chinese is tricky because everyone has their own way to express "certain sentences". To summarize, 下初雪 = 下第一場雪, 初雪 is more formal since 初 means "first", "beginning" or "commencement". On the other hand 第一場雪 means literally "first snow", 場 is the classifier for a "snowstorm", "blizzard" or just "snow" in general. It is more redundant, hence it is colloquial. Usually, but not always, concise phrases tend to be formal since colloquial phrases are typically long and redundant. (And sorry for the delayed reply.)

Thank you, Kaj. Even after the verb 下, are you (personally) more likely to say 下初雪 in conversations? Thanks again.

I totally agree with Kaj