The word なし, of course, means ない, and it is defined as such in dictionaries. But why does this word exist? Are there even any situations where you can say なし but you can't say ない? Is it a remnant of something from classical Japanese? It seems similar enough in use and meaning to ない that it doesn't seem that it can be dismissed as its own word entirely and etymologically separate from ない. I checked etymology sites and dictionaries and tried googling, but I can't find any explanation for this word.
nasi and nai are the same word. Like all adjectives, nasi is the conclusive form (終止形), while nai is the attributive form (連体形). More specifically, the attributive ends in naki, but the medial -k- drops out in modern Japanese becoming nai. This is true of all adjectives: atusi -> atuki > atui, takasi -> takaki > takai, muzukasi -> muzukasiki > muzukasii etc. You can still see this medial -k- in the adverbial form (連用形) -ku.
Note that in modern Japanese, the original conclusive was replaced by the attributive, so it may now act as a conclusive. This is true in verbs as well.