In casual conversation, the two words are used almost interchangeably. In a court of law, with a judge and lawyers and stuff, there's an important difference. But if I'm talking to my friends, I will use either term and mix them up in the same statement even. So, yes, your PS is correct, but native speakers are probably not making that distinction when they speak.
A leap is more graceful and poetic than a jump. A newspaper will report that a woman "jumped." A novelist will write that the same woman "leapt."