This is a long-term, persistent problem.
Firstly, you must have a hardware calibrated monitor or your grading will not be done to a standard. For example, if you monitor has a blue cast, like most, you will overcompensate with warmer colours to get the 'right' look. On a calibrated monitor, your graded video will look too warm.
Once you know that your grade is to a calibrated standard, there is still the video sharing site colour shift issue. The colour shifts from what you see in FCPX to be more red and more saturated, as well as darker. I suspect this happens with all video sharing sites, but I know it does happen with YouTube and Vimeo.
I suspect this happens when the sites convert to Flash for streaming. It happens when I create Flash e-Books from my photographs, which are immaculately graded (and even when played back on my own computer). If you download an MP4 from Vimeo, for example, the colour looks correct while the streamed version does not. There is a similar colour and value shift with profile photos on LinkedIn as well as thumbnail photos on YouTube.
Each publishing media requires it's own grade. Whenever I publish to Vimeo or YouTube, I make a final graded master and then I make a special web master. The web master is a copy of the master that is graded again to reduce the saturation and red component, and lightened. Some trial and error is required because YouTube, for one, seem to vary the colour shift from time to time. I will also create a special broadcast master if I'm preparing a video for network broadcast, which has extremely strict colour and luminance standards.
I can't say why the latest iPads read the colour correctly. Maybe Apple is compensating for Flash streaming.
I'm not sure how you had a clip with the Adobe profile embedded open in FCPX with the same profile, unless you were working on a photo and saw the profile in a photo editor, which you later imported into FCPX.
When you create a HD video project in FCPX, it automatically assumes a Rec. 709 ICC colour profile. If you import other assets, like photographs, that have another profile embedded, FCPX coverts to Rec. 709. When you export, that's the profile it uses. You can check if your video has a profile embedded by using 'get info' in the finder. A HD colour profile (Rec. 709) will be shown as HD (1-1-1).
