The description is misleading. But overall does what I need.
This is NOT correct. BUT, you just paste in the Tag Manager JS and you are good to go. The other plugins I found (which DO just need the GTM ID) all seem to put it in the footer (whether they say that or not, I looked at the source code of a few).
As a added bonus, it can insert the Google Webmaster Tools (and Bing) verification tag into the site as well. I had trouble getting GWT to verify with the verification tag via Google Tag Manager, but this should work fine.