I found the box, ref C2S45579 written on the UROPARTS label, so I believe it’s the correct one

 

Are you triple dog dare positive the part itself matches the label on the box?

It is rare, but sometimes the wrong part gets put in the box. Not necessarily at the manufacturer, but possibly at the vendor.

 

I can’t find anything about URO-024305, I will call the shop to verify.
Usually they have manufacturer docs, hopefully they have some for the tensioner.

Also they can unbox one of them from stocks

 

I, too, can't find anything about URO 24305... - which makes me think that maybe that is not the URO P/N of the tensioner, but only of a sub-part of the tensioner, and no-one except for URO would know about that number. But it's worth double-checking with the place, where you bought it, that that is what is supposed to be in the C2S45579 box.... - maybe compare a part they have in their box with your part. Also check, if it is in any may possible to mount the pulley on that tensioner incorrectly... - compare that with the part in the shop as well.

PS:
Actually: Looking at your build in tensioner (where it is utterly misaligned with the belt) and looking at the pictures, where your old and new tensioner are side my side...:
Isn't the pulley mounted on the URO part the wrong way 'round? Be careful, if you want to open that bolt: It most probably opens by turning it CLOCKWISE!

 

I already installed it incorrectly once and it’s visible, and more when tightening the bolt.

On the product page, the pulley is also mounted like this, so I think it’s OK.

On the old pulley, i had to arm the spring by pulling off the tiny metal part. I didn’t find this on the new one, and as the angle is simular, I assume there is none

I will mount the new one and try to measure the angles. Also I will inspect the holes where seat lugs

 

I am not sure, if you understood, what I meant exactly...
I have copied in 2 links above for that C2S45579 part - strangely, when comparing those 2 links, one shows the pulley one way round, and the other link shows the pulley the other way round - that link, where I think the tensioner is shown with the pulley mounted correctly is:
https://parts.jaguarlandroverclassic...tensioner.html

Because if I look at the picture of your belt NOT sitting correctly on the pulley of your tensioner suggests that the pulley needs to come further away from the tensioner-arm it sits on.
Can you try to turn the pulley around and check, if the pulley then sticks out further away from the pulley-arm on which it sits?

OR:
A totally different approach of thinking...:
I think it is safe to say that SOMEHOW the pulley needs to come further away from its tensioner-arm.
Maybe the problem is as simple as this:
Maybe the aftermarket manufacturer URO simply is lacking quality-control and as a result of that they are assembling the tensioner incorrectly.
I am confident that I could "fix" your part, if I had it here:
And if you have an hydraulic press, you should be able to fix it yourself - although you should not have to, because the shop, who sold you that part should fix it for you:
It should be possible to simply shift the bearing inside of the pulley into the correct position so that the pulley ends up in a position further away from the tensioner arm when mounted.
Just try, if the pulley needs to sit that way or the other way, and shift the bearing inside of the pulley (before mounting the pulley) to the CORRECT position.
This should solve your problem.

 

I think the scheme is wrong, why ?

The tensioner looks like it is XR86445, it’s the old tensioner in my case.

Also, if you look at the price, just above you have the number of the part on the scheme.
On the scheme, the tensioner is (6), however the product says it’s (4) on the scheme

 

I had problems understanding what you meant with scheme - but I think I just figured it out:
I think you are referring to the illustration (=scheme!?) shown in:
https://parts.jaguarlandroverclassic...tensioner.html

Yes, you might be right that they made a mistake in that illustration. C2S45579 is not part 4, but it's part 5, but that does not change anything.
C2S45579 is still the tensioner.

And it also does not change anything I wrote above about the pulley probably being mounted the wrong way round or the bearing in the pulley being positioned incorrectly or BOTH.

And what now about XR86445? Do you think that that is the part, which you have now from URO or do you think that this is the part, which you need?

 

The scheme is from XR86645 page. That’s what I meant : https://parts.jaguarlandroverclassic...tensioner.html

XR86645 is the tensioner for S-type from 1999 to 2000.5 and C2S45579 is for S-type from 2000.5.
XR86645 is OEM tensioner and it’s the tensioner on the right on my pictures. It used to have this ref : XW4E-6B209-AD


I can try to swap the tensioner pulleys, also need to check pulley angles for crankshaft and idle pulley (1).

 

OK, I see - there are 2 illustrations, both are identical, even though one is for XR86645 and one for C2S45579.
You have the latter one, and even though most links will not tell you that, you need XR86645.

But I cannot see a difference in the parts, so it probably comes down to what I wrote a few times before:
The pulley on the tensioner is probably being mounted the wrong way round or the bearing in the pulley being positioned incorrectly or BOTH (that being the difference between XR86645 and C2S45579).