Hey, @Tryoxis
First off, I'm sorry to hear about the trouble you're suddenly having with the game since the latest update. It's very strange though that the disc is able to work on a different Xbox and not yours. To leads me to immediately believe something may be faulty with your Xbox's disc drive with it having trouble reading that specific disc.
I don't know if there is an option for you to try out another disc whether it be a friends or a rental version, but if another disc version of the same game works in your disc drive I think it would be safe to assume it's faulty disc. Are there scratches on the disc or was there any permanent marker/pen marks on the top of the disc? Some people wrote on discs in the past but now that can cause issues with the laser reading it.
Even though you have automatic updates enabled, would you mind going to your Xbox Settings and just double-check to verify you have it fully updated?