https://blog.fcpeuro.com/m52tu-m54-s...elts-explained
^^^ that's actually a nice resource there, clearly showing the belt path, etc.
They used both mechanical (spring) and hydraulic tensioners on these M54 engines. Can be replaced interchangeably.
My 2001 has a mechanical one (and came with one). From @Russianblue's video, it seems to be moving as my mechanical one would. I don't know if they switched over at some model year (2002?), but even if I did, the fact that they can be replaced with either one means it's probably unknown.
And here's the idler pulley.
https://www.fcpeuro.com/products/bmw...ey-11287841228
That little nub sticking out on the back side is supposed to fit in a hole to keep it aligned. That's the thing that is pretty easy to get wrong.