Word 2013 can quite happily word with documents created in Word 2002. There is nothing to install; neither is there any need to change the save format in Word 2013.

Whatever issues you're having vis-à-vis the Track Changes display has to do with how the app is configured. That's because how tracked changes are displayed depends not on how they appeared on your system when creating them but on how the person viewing them has their system configured. The same issues can arise even between different installations of Word 2002.

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Cheers,
Paul Edstein
[Fmr MS MVP - Word]