I saw the Death Note musical on stage in the September of 2023 (of which more here), and one thing I thought was interesting was how different Light and Ryuk's dynamic felt, compared to the manga and anime.
(NB: the only version of the musical I've seen is the English concert version, in which the dialogue is cut down slightly, so I don't know whether the Light-Ryuk relationship has significant differences in the full English version of the musical (fingers crossed we get a full version eventually!), or in versions in other languages.)
In the original Death Note, Light treats Ryuk as a pet and confidant. They're not friends, exactly, but Light does seem to find something of value in Ryuk's company. Being Kira is a strange and isolating situation, but Ryuk's presence does a lot to combat that isolation; he's the only 'person' Light doesn't have to lie to.
In the manga, when there are cameras installed in Light's room, Ryuk says, 'It might be real hard to go without apples, and even harder to be in your room without talking to you.' I think it's very cute that Ryuk's biggest concern here is not getting to chat to Light. It must be frustrating for Light, too, not being able to talk about his plans with anyone; how is anyone supposed to admire how smart he is?
It's also cute that, as soon as Light hears the hidden cameras have been removed, the first thing he does is go to the kitchen and get Ryuk an apple; it suggests to me that Light really does have some sort of fondness for Ryuk. They even canonically play videogames together in their spare time!
In the musical, Ryuk is an annoyance who cannot be repressed or controlled. And it's hilarious! He pulls Light towards him by his tie; he puts his arms around Light from behind. He ruffles Light's hair at one point, and Light is so put out. Musical Ryuk is a constant mildly annoying presence in Light's life, providing obnoxious commentary and patronising little touches, and it's incredible. But I love their dynamic in the original series as well; anime/manga Light views Ryuk with something almost approaching warmth, and warmth is hard to come by when you're dealing with Light Yagami. Musical Light would be happier without Ryuk there, whereas original Light would be sorry to lose him, although I doubt he'd ever admit it.
Another distinction is that Light thinks of himself as outranking Ryuk in the original series, and will occasionally scold him or use him as a tool - persuading him to search for cameras, for example - whereas Light clearly recognises that Ryuk is something completely beyond his control in the musical.
I suppose the distinction can be summed up as 'in the original series, Ryuk is Light's pet; in the musical, Light is Ryuk's pet'. Light is, perhaps surprisingly, a more considerate pet owner! He's patronising, but, when it won't blow his cover, he's always willing to indulge his pet shinigami's questions or his desire for apples. Musical Ryuk, meanwhile, just constantly affectionately harasses his pet human for his own entertainment.
They're both interesting dynamics, in any case! I think the relationship between Light and Ryuk is often an underappreciated aspect of Death Note.