Make sure the category you're change the settings in does not inherit the params of another category.

Related items are retrieved based on tags. Are you using tags for your content in K2?

Can you paste the contents of your item.php please?

If you don't see the "introtext" in related items, it's a configuration problem and you need to dig deeper in your category settings. Also make sure that the related items are not explicitly hidden in the item display settings, I've seen this in the past happen accidentally by users.

The default code as is works just fine. To verify, simply create a new Joomla installation, add K2 in it and either create a few dummy items/categories or use our own "Fill It Up" (github.com/joomlaworks/fill-it-up-for-joomla) to quickly generate dummy data in K2. Then test the settings again by allowing related items' introtext to be shown in the display settings of one or more categories.

Nope, perhaps your if statements got mixed up in the overrides. Simply place an underscore in front of "com_k2" in your template's /html folder so the default templates kick in. Do you still see the issue?

This boils down to one thing then. Make sure there is no inheritance issues in the category parameters and/or menu items and item parameters directly.

I can also verify that I cannot reproduce your issue. Then again, my testing installation is mainly stock K2 settings.