This was already addressed through a support ticket and found to not be a problem, but just the wrong use of the event ID. Also ranail checked out our video tutorials after posting this and found his answers
For anyone else interested...
Hover over the registration link for the event in the main list of events for DT Register. You will see the url in the bottom of your browser window. In that url is "eventid=xx". The number you see there in place of the "xx" is the event ID. This is the number to use in the plugin code.
So on your page, you can insert:
{dtregister}87{/dtregister}
...and a registration link (or button depending on your settings) will show for the event with ID #87.
If you assign an article to an event in Event Management, the registration buttons will automatically show at the bottom of the article. If you manually insert this plugin code somewhere else on the page, the buttons will NOT show on the bottom still.
Of course, you need to have the EventLink Plugin installed AND enabled for any of this to work. Don't forget to check out the video tutorial that shows how to setup and use this plugin.