Logan is toxic. Both in game situations and when you bring it to the real world.
He is toxic because he knows how stubborn he is, but he don't do a thing to change it. It just happened because the MC appeared on his life.
And you can say: "But he is hurten and don't know how to call for help or is afraid or it's too prideful to do it" and you're right.
The fact that he is a toxic person does not mean he isn't a human being. He is suffering A LOT through and before game events (since his marriage was basically dead).
On my opinion, what happens to Logan is something similiar that happens in psicology, during a relationship between a psycologist and a pacient. It's very common to the pacient to do something we call "emotional dump" (I don't know the proper word for it in English, sorry) where the pacient start to get feelings for his/her psycologist.
This happens because the psycologist figure is the one that makes you feel better. That understands you. You are not in love with him/her because you like him/her, but simply because it makes you feel better about yourself and you want this person closer to you.
This is basically how I ser LoganXMC relationship. You are the only one trying to help him regardless of how rude he treats you. You are the only one who saw him at his worst and stayed by his side. You are the only one who he feels comfortable to talk to. You make him feel better about himself, you make him feel that he is not a monster (despite he totally acts like one and it's totally understandable why people think what they think about him).