Gravity assist or gravitational slingshot [1] is a specific type of maneuver that's different from the Hohmann transfer that Chandrayaan-3 is seen taking in your video.

[1] https://en.wikipedia.org/wiki/Gravity_assist

yeah but notice the JAXA one is different, since it approaches the moon twice.

It starts sort of the same way, but then it uses the moon's own attraction to project itself much further out and then come back approaching the moon in a different way.