DOM is separate from Ecmascript, but is actually one of the 3 components of Javascript (along with Ecmascript and the BOM). This is at least according to Professional Javascript for Web Developers (you can see it on 'page 3' in the Amazon preview).
I would say that what node.js runs is ecmascript with it's own API that differs from the DOM. It's called javascript for convenience, but then what separates javascript from ecmascript?
Nothing. It is only different in name. Oracle owns the actual copyright on JavaScript via Sun. ECMAscript was used so the ECMA would not have to worry about this.
I guess this is a point of contention. Javascript is considered by many to be an implementation of ECMAscript that also provides additional features (the additional features, according to the book I cited, being the DOM and BOM).
Looks like Javascript IS the Assembly of the Web.