Do we pray to save our souls (SOS)? Or do we pray for us getting anything and everything along with moksha? Latter.
Herein lies main difference.
Eg. gajéndramoksha. When divine intervention happens in such instances, it is usually via some agent or direct.
But will you call shri Hari saving Gajendra from crocodile as him taking an avatāra?
If there is no intervention, we accept it as our Sanchita-karma and move on with strife.
Because we (ancestors, previous births) did not do our karma and did not upheld the dharma by doing the right thing.
If citizens of mathurā had rebelled and overthrown Kansa after his adhārmik usurping of power, will shri Hari need to come down?
Because there are some things more valuable than existence of Srishti which needs to be protected. It is to save that thing.
It actually means to save that/them which/who is/are so exalted and sacred that its worth saving at all costs. Even if that cost is pralaya. Or world war. Or systemic reset.
But all of those who died then were ill-fated (due to their past bad karma) to live in such adhaarmik times where even the queens are disrobed in public and suffer the consequence of that age and times.
All those who got to see hear touch Krishna in Mathura and Vrindavan must have gotten good gati and it was as a result of some powerful accumulated puNya.
Think of how many murders Putanā committed before meeting her end at hands of Krishna?
But unfortunate enough to lose their new born babies? Do you call it a good fortune? 😊
मा ते संगोस्तु अकर्मणिं..... Never indulge in Akarma (dodging one’s duties and responsibility towards dharma).