Kristian,
Sounds like theres a difference inthe preferences preset in each computers post processing programme. All you can do is open the preferences and very carefully copy all the various settings in one, then do the same in the other and note any odd differences.
My standards for Photoshop 7 are:
RGB Working Space: AdobeRGB. sRGB is a very common alternative.
Color Management Policies: Preserve Embedded Policies
All profile mismatch and missing profile dialogues activated.
Conversion Options: Adobe (ACE). with either Perceptual or Relative Colormetric as the intent.
Black Point Compensation and Dither both activated
I always shoot in RAW and try to stick to AdobeRGB unless I know I need to use sRGB for a specific reason instead. I've never identified a visible difference in prints made from each of these, so we are nit picking.
What post processing software do you use on these two computers? The same, or two totally different software packages?
Colour temp 6500 and a gamma of 2.2 with a luminence of 120 is considered "Normal", but I use a luminence of 110, and at least one user here reports using 90 with great success. Forget about any shadow, or black luminence. This will be impossible to alter and will send you off on a wild goose chase that WILL screw up everything else.
The 6500 colour temperature should give your screen a very slight pink tinge at first sight. Some years ago a large paint manufacturer here in the UK produced a range of white paints with names like "Harvest White", "Blossom White", "Rose White", all whites to the eye, but different when held against each other.
I have never used your specific calibrator, but I'd be surprised if it cannot deal with the issues. So don't give up on it just yet.