How should I connect my new Fagor wall oven to the existing wiring?
This is a great example of color codes not meaning that much in North American wiring. 

Ground color codes are very clear -- ground is always green, green/yellow or bare. Other colors not so much though. 

The nameplate on the oven is extremely clear: It is made for North America (60Hz rating, highly relevant to the clock).  It is made for 208-240V, which means aside from ground, it needs two hots. That white wire is a hot, and should be marked with black tape to indicate it as such.   It is not a 120/240V appliance, which means it does not need neutral. 

The white wire in your wall is neutral, and is useless for this oven (but useful for most other ovens). Cap it off. I certainly can't tell you to mark it with green tape on both ends and open up your panel and move it from the neutral bar to the ground bar, because remarking wires to be grounds is not allowed unless the wire is #4 or larger. 

While it is "technically leeegal" to use a neutral wire as a ground wire on a 120V/240V appliance, it is not for a 240V-only appliance.  Therefore I would aim for the best of all worlds, and retrofit a ground wire from the oven location to the panel the oven is powered from.  #12 will suffice. Bare wire will suffice, otherwise it must be green or green/yellow.  The stuff in the wall isn't MC, it looks to me like flexible metal conduit - and it's possible (easy?) to push/pull additional wires into conduit. 

On ovens, "which hot is which" doesn't matter.  It's perfectly fine to hook up two hots (black and black-taped-white) to two hots (black and red).