Just to follow up to this discussion, yes, you have to copy the settings for all these extra app modes separately. That goes for Game mode as well if you play video games. As far as I could find, there's no way to automatically do it.
Fortunately, color temperature covers the most important thing and does carry over, but you'll still have to manually copy over Picture settings -> Advanced settings -> Per Color Adjustments (6 colors, 3 settings each) for each mode for some reason (unless this has changed in a recent software update; I haven't recalibrated for a few versions). I believe I had to copy these over four times: Apps, Apps (Video), Netflix app, and Game Mode. [Because of the sound-issues, I'm using almost all 3rd party apps via AppleTV or Roku now where possible, so it's not necessary if you use those boxes of course...still will need to for Game mode for game consoles, though]. And yes, you have to make sure brightness, all local dimming or enhanced picture settings, etc. are matched up too with your other modes. These are all treated separately for those modes I believe.