What is a Kill?

If you knock down the enemy you always will get the kill no matter how much damage you did. So last hit = knock down = kill, except you get killed yourself while your victim is knocked down, then the one who kills / finishes him will get the kill count. 

Finish moves will not give you the kill if someone else knocked the enemey down. 

You will get an assist always as soon as you did damage to the killed enemy no matter how much damage.

So get your * hands off of the knocked down people, they won´t get you kills / the kills can´t be stolen from you. Just ignore them and focus on the left overs who are still able to fight back or revive them. No reason to try kill stealing in this game... not possible.

Conclusion: If you knock down - you get the kill except you get killed yourself, if you do damage in the fight but don´t knock them you will get an assist, if you damage a knocked down enemy or kill a knocked down enemy you didn´t knock down yourself, you will get an assist (assists are nowhere to be shown so ignore assists as well, you can not show them off)

Edit: I think it´s needless to say that not every person you knock down will also die, because some people will get revived. So yeah, sometimes you deal tons of damage but don´t get the last hit or people get revived after you knocked them... can be frustrating but also very rewarding if you get in a fight with 3-4 teams,  knock down a few people in the fight and then when the last get killed you suddenly see a few kills pop up in one second :D 

Edit 2: Not sure what happens if an enemy knocks down another enemy and you finish it. Not 100% sure if you would get the kill or the enemy. But in your own squad you can not killsteal. So you always will get the kill if you knocked down and your squad mate finishes or the other way around.