Word-Request – Term for a Cover with Changed Lyrics

This is called a lining:

material that lines or that is used to line especially the inner surface of something (as a garment)^(MW)

You would ask questions such as:

What are those boots lined with?

Is the lining removable?

Do you have any gloves with fur lining?

It's a montage:

[Merriam-Webster]
1 : the production of a rapid succession of images in a motion picture to illustrate an association of ideas
2 a : a literary, musical, or artistic composite of juxtaposed more or less heterogeneous elements
2 b : a composite picture made by combining several separate pictures
3 : a heterogeneous mixture : JUMBLE
// a montage of emotions
// a montage of sounds

Although the first definition indicates an association of ideas, you can still have a montage of things that are unrelated to each other, but which still happen in the context of something that unifies them.

For instance, a quick series of completely unrelated images that all occur because of having taken drugs or having a confusing dream (or anything similar). It's the altered perception that provides a common contextual theme—as shown through a montage of imagery.

There is also another technique that isn't so much about disjointed images, but about shortening a lengthy event by abruptly transitioning from one part of it to another—losing what happened between them. It can happen once, or it can happen multiple times in sequence. The sudden transition is called a jump cut:

: a sudden often jarring cut from one shot or scene to another without intervening devices (such as fade-outs) broadly : an abrupt transition (as in a narrative)
// So there is less need for all those annoying rapid jump cuts that detracted so much from the choreography in the first season.
— Jennifer Ouellette, Ars Technica, "Iron Fist’s second Netflix season mostly lives up to its promising trailer," 10 Sep. 2018

If I'm not mistaken, it was the Bourne movies that really started (or perhaps just exemplified) the trend of both shaky cameras and a series of jump cuts with fight choreography. But, the shaky cameras aside, jump cuts have been used for a long time. The idea of shortening time by only showing key elements of action (not just fighting) can also be represented through a similar series of very quick fade-outs and fade-ins.