You are right about your understanding of grammar as far as modern conversational Japanese is concerned. Here the に is used as an attempt to sound poetic and sophisticated and it means "..の中で" or "...という場所で". This use of に is not so uncommon in literary language, even now. For example, you could say 沙漠に倒れる and it means something like "perish in the desert" (notice the choice of the word perish rather than die). 光に舞う粒子 is particles dancing in the light. Also, Thomas Mann's novella "Death in Venice" (or Der Tod in Venedig if you will) is known in Japanese as ベニスに死す.

Today virtually no one speaks in that way, but you may find this usage of 'に' in old movies or books. It is a literary dialect. The 'に' does not mean 'to the rain' but means 'in the rain'. Examples: 駅前に待つ 母校に会す 会館に昼食する

これは「雨の日に」の意味ですね！

The direct translation of "Singing In The Rain" is 雨の中で歌っている/ 唄っている but it's not beautiful nor poetic. When someone translates titles, songs, sentences, books, etc, or write subtitles of the movies, they need to change the meanings to make them beautiful, understandable and natural. So, there are many cases that titles of songs, movies and books are totally changed.