What does "主体责任" mean?
I believe 主体 here means "subject" or "agent". The individual (main) body, be it a natural or a juridical person. 

The descriptions of this principle that you can find on Baidu Baike stress the fact that responsibility must be carried by all party organizations（各级党组织）, that responsibility must be capillar （细化责任）and that it must trickle from the top down to all levels （以上率下，层层传导压力，级级落实责任）. 

It makes sense in the anti-corruption campaign. It's not hard to see how corruption can be effectively fought only if everyone at all levels is accountable within their own jurisdiction （种好自己的“责任田”, i.e. they must sow their "responsibility field"）. 

And this can be done if they all really bear their own load （真正把担子担起来）.

So given all this, it's not very easy to translate this concept into English, but I think we can agree that the one you mention appears inaccurate.

I would translate this as "agentive responsibility", or, more simple, "subjective responsibility".