헤어지다 covers both saying goodbye for the day and ending a relationship, and the -게 adverb says in what spirit it is done. 쿨하게 헤어지다, appearing twice here, is parting without a fuss; 너무 무겁게 헤어지는 것도 아니고 is not making the goodbye too heavy. The break-up sense is the one behind 어떻게 헤어지든 이별은 아픈 법, 'however you split up, it hurts'.