I play Overwatch and I've been wondering about this for a bit. 

When the team loses, and quickly at that....how can you tell if it was caused by your own actions versus those of others. 

I wish to figure out how to better myself, and that in turn requires me to figure out when I make the major mistake, versus someone else. 

If you have to ask it probably was you.

I'm gonna try a very general answer since I don't know what role you usually play:

What I try to do personally is after every death I ask myself "Could I have avoided dying there?" Or least "Could I have died later in the team fight?" Often times the answer is yes.

Then it's a matter of figuring out what you could have done differently. I play support a lot and sometimes my teammates do stupid shit (like Reinhardt charging into 6 people) and then I do something even worst by thinking "I'm going to save him!" which of course gets me killed.

With replays now, you can auto-review your games pretty well. Checking out things from the enemy team's point of view and figuring out if you're out of position in certain situations, if you lack awareness, if you used an ult when the fight was already won, etc...

How many messages do you get reminding you about sexual relations the sender had with your mother?

Thats how I judge whether I contributed or not.