Yes, both are playing Lee Shaw but the Apple TV+ website specifically states that Wyat Russell's character is "Young Lee Shaw," just like how Anders Holm's character is "Young Bill Randa"- https://ibb.co/wgYWvDT.

Shouldn't the on-screen credit have precedence over any other sources? Because it definitely passed more stages of approval, and can't be easily changed.

IMO... This specific on-screen credit is designed this way so they can transition between the two Russell's with the Shaw character across the time lines on the same screen. If you watch the credits itself you'll see this which doesn't come across in the screenshot provided above.

Using "Young" makes sense especial due to the fact it's called out that way online.

That's not the point. The point is that in some situations the "extended" character names taken from other official sources (website etc.) are being re-edited by the mod to the way they are spelled on-screen. Stripping last names, ranks, or whatever. While in other situations the same mod prefers to use the names from other sources over the on-screen ones. That looks totally random, more like a personal preference, not the question of correctness, logic, or following the rules. Hence the question.

I understand and have seen the discussion here -> https://www.themoviedb.org/talk/655739f053866e00ff07ebe3 but in this example IMO (which is why I started my comment with it) the use of Young seems correct.