"The intermix chamber was never pointed out in the movie".
Sorry, I don't get your point here. Are you saying that because no character pointed at the two big glowing shafts located across several decks and monitored from main engineering and said "that's the intermix chamber, you know", that we don't know for sure if it is the intermix chamber? I think that's a little silly.
The central crystal chamber was never identified, though - that's correct. We don't know exactly where the crystals were located in the refit ship, or exactly what purpose that reactor room served. However it is logical to infer that the room does house the crystals when we see the similar piece of equipment in The Voyage Home.
"YET we know that the crystals mediate the reaction of matter and anti-matter so why would the chamber be in another room?"
Quite. It's another design element introduced in the second movie that doesn't really make any sense and was clearly added for the narrative need to provide Spock with a "sealed" room in which to venture to fix the warp drive and succumb to radiation poisoning. My theory is that perhaps matter and anti-matter are first introduced to the Crystals in the reactor room and the resultant plasma is siphoned away into the intermix chamber where it is further 'processed' in some way. But it is admittedly a flimsy theory based on a supposition, caused by a frankly baffling design detail.
The schematic you've provided there actually raises more questions (and is, I believe, not produced by official sources but by a fan). For one thing, although the intermix shaft connects with the deflection crystal, it also seems to be bypassed by what looks like another horizontal intermix shaft, albeit a short one. Which raises the question - if there is a second horizontal shaft which connects directly to the impulse engines, why is there a need for the deflection crystal at all? It's also interesting to note that there is no area for deuterium storage specified - unless the area at the bottom of the vertical shaft contains both matter and anti-matter, which would be acceptable as this M/ARA system does not appear to operate using the "injected at opposite ends" method (hence the lack of a centrally-positioned crystal in the shaft itself).