"CliveHarison. Why do we need any other words other than Rise. Fall. Drive, Swing, Sway."
Because those are specific actions.
"Lilt" was used to represent a more general property, which can be satisfied in a number of ways, including BUT NOT LIMTED to combinations of the terms you mention.
"I can't imagine any teacher saying I would like to see you lilt on that step."
I could very well see a teacher asking a student to create a more lilting quality in their waltz.
That's a general request. They might then tell you to do some specific things in order to achieve it. For those specific things we have other words - but the other words that you've proposed do not capture the general effect, as they only represent some of the many ways in which it can be achieved.