Provided you have access to the 2010 media and its product key, for emergencies, you should be able to simply uninstall the trial version without affecting 2010, as they occupy different spaces, though they share some resources. It will then be necessary to repair 2010 in order to reset the links you refer to.

You can probably leave it in place and re-register 2016 (see Use different Word versions together ) but the chances are that updates will reclaim the settings for the later version.

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Graham Mayor - MS MVP (Word) (2002-2019)
Visit my web site for more programming tips and ready made processes www.gmayor.com