Assuming the original is European oak, then where it grew is immaterial because it will almost certainly be either Quercus robur or Quercus petraea, although there's an extremely slight possibility it might be Q sessiliflora or Q pendunculata. If a buyer comes across a description of oak that includes a place name, such as German, Scottish, English or French preceding the name 'oak' it means only that's where the tree grew, but it will still be one of the species I've listed. The fact that boards from a Quercus robur grown in England might have somewhat different physical characteristics to boards from a Quercus robur grown in Germany is entirely down to different growing conditions, e.g., soil type, climate, open grown, forest grown, altitude, etc. There are several species native to North America that fall into the general categorisation of American white oak, principally Quercus alba, Quercus prinus, etc. For your project Rorton, I'd use whichever of all these oaks that came at an acceptable price because in a finished piece they will all come out as acceptable.
On an unrelated side note, I have come across quite a number of people that swear they can readily distinguish American white oak from European oak just through a superficial inspection, i.e., they just look at it. However, whenever I've had chance to put under their nose a few known (to me) samples of each type, I've never come across anyone that reliably can tell one from the other.
The stain/dye sample is a pretty good match, but it's a little bit red compared to the original. If it's important to you to get a closer match you can reduce the redness by adding a little green dye to the mix. Just make sure you use a dye with the same solvent, which I think, as you say, is white spirit. Check the can of the stuff you already have to find out what the solvent is.
Fuming is a technique I'd avoid, one reason being that you can end up with very different results from one board to another, especially if your oak comes from different trees, which is highly likely if you buy wood from a large pack. And, of course, you need a set-up to do it, ideally including a fume filtering respirator; setting up for fuming is not especially difficult, but maybe more faff than you really want to take on. Slainte.