OK....so red oak has a cinnamon tone with plenty of red-gold in it. Anything that is "light" will HIGHLIGHT the red. So....if you want to kill the red you MUST work with a mid-tone floor or DARKER. Right. OK. Now what?
Your professional flooring installer will offer colour patches directly applied to your installed floors. I suggest working with ANY colour that has green undertones in it. It's pretty easy to figure out. Put a drop of the stain on a white piece of paper and smear it around. If the colour seems to have a bit of green to it you are in the ball park. Avoid ANYTHING that is orange, gold, red or red-brown.
Once the flooring professional has applied 4-6 different stain patches, s/he will apply a single coat of the finish over top (you MUST see the colour with the finish). And be aware that solvent based finishes turn ORANGE (again...must be avoided). Water based finishes will offer a clear finish without discolouration.
Now that you have the stain patches, go ahead and pick the one that is as close as possible to your favourite. Be aware that your inspiration floor will not be achieved. You can come close...but it will be 2 steps darker than the floors in your photo. That's OK. The "tones" are the important part, not necessarily the depth of colour.
Q
Hand scraped hickory with gray wash
Q