You are right, both 红书 and 红色的书 are technically "red book." However, we often preferred one over another in a given situation. In general, for a Chinese learner to talk about the color of an object, using X色的 is safer and prevents ambiguity. I will go with 红色的书 myself.
A lot of colors could be used directly without 色的 before a noun, and either way is perfectly fine. Here are some examples:
红衬衫 = 红色的衬衫 = red shirt
黑猫 = 黑色的猫 = black cat
蓝笔 = 蓝色的笔 = blue pen
In these examples, as a native speaker, I do not have any preference over one or the other. (I would choose the shorter one if I am writing, just to save some time lol)
However, sometimes using the color word without the 色的 feels too "specific." With certain words, it just feel like it is a particular name of something. For example, 红书 for me sounds like "THE red book" instead of a red book. As I looked up online, <红书> is a translated title of a book by Carl Jung (The Red Book.) There is also an application on market named 小红书 in China. Therefore, I would prefer to use 红色的书 to avoid being ambiguous.
In poetry and, as you just mentioned, songs, we often use stylized language to make things stylish, pretty or just to rhyme better. The lyricist of the song you mentioned probably felt 红 is a better fit than 红色的 in the song.
There are also cases in which X色的 makes a major difference:
白色的包: a white bag
白包: a white envelop filled with money given to the family of the deceased in a funeral
黄色的小书: a small yellow book
小黄书: pornographic book
Some nouns with color words comes only in one form:
白板: white board (not *白色的板）
黑人: black person (not *黑色的人)
绿茶: green tea (not *绿色的茶, unless you are talking about a cup of tea that looks aggressively green! It sounds like poison!)
As you can see, using the color word without 色的 could mean something very specific sometimes. This is the main reason why the book recommended that you should use X色的. However, as you continue to study Chinese, you will gradually get a sense of whether you can drop 色的 naturally in context :)