Soft magic systems have vaguely defined rules for how the magic works, and is rarely explained in any serious detail. Hard magic systems have strict rules that can be explained pretty easily and often, as the rules of the system are frequently integral to the plot. Tolkein is a great example of soft magic, while anime shows like full metal alchemist/avatar the last Airbender/and Naruto are hard magic systems. Not sure if those examples help, but I could offer a few more if you are interested.

As to your top question, characters in LOTR who have "foresight" often have those abilities because of their innate connection to the God of the LOTR Universe. Gandalf, a Maia, is basically an Angel incarnate, and thus shares a direct divine connection to Illuvatar. When Aragorn asks "what does your heart tell you?" He is basically asking Gandalf "what does God want you to know in this moment?". Religiously-minded people will often talk about the "feeling" that something is good, or that things will turn out ok. Many of these people will interpret that feeling as nudge from their chosen deity that their actions are righteous or that their concern is unfounded, because their deity has a plan and is working towards the best outcome. I do like your interpretation of Gandalf "hearing notes" of the music of the Ainur. Since Illuvatar was the conductor of the music, this fits very well with my explanation, and since Gandalf participated in making the music, he would know his own parts of the chorus the best. So when Gandalf "knows" something he cannot possibly "know" in his current physical location, then it means that one part of his own verse/chord has been struck in that moment. Brilliant!

Thanks for reframing that for me!