I am an American citizen, looking to fly to the EU. I am feeling I do not need an ESTA because the wording implies the necessity of the document belongs to non-US citizens. May be an obvious question, but I would like to be sure. I wouldn't want to ruin plans due to a misstep.
You are right. The ESTA is needed only for citizens of certain non-US countries who want to visit the US under the Visa Waiver Program.
The US doesn't require authorization, permission or registration for anyone trying to leave the country. And as a US citizen, you also don't need permission to come back, just your passport.
(When you fly back to the US, there is a process called APIS by which your airline will let the US government know who is coming. So your airline will ask for your passport details at some point before your return flight, either at check-in or online. But other than that, no action is required from you.)