You ITprimer seems to disagree with your statement:

Emergence is NOT the sum of the parts.

ITprimer:

(3) the non-trivial interactions result in internal constraints, leading to symmetry breaking in the behaviour of the individual components, from which coordinated global behaviour arises;

(4) the system is now more organised than it was before; since no central director nor any explicit instruction template was followed, we say that the system has ‘self-organised’ ;

(5) this coordination can express itself as patterns detectable by an external observer or as structures that convey new properties to the systems itself. New behaviours ‘emerge’ from the system;

Non-trivial interactions of individual components -> Self organization -> New behaviors labeled to have 'emerged'

Where did they emerge from? The non-trivial interactions. This description runs counter to your discription "Emergence is NOT the sum of the parts." It is the sum of the non-trivial parts by the above description and a loose definition of sum.