I'm a bit uncertain by what the question means exactly by "distinguish." I believe that the question is asking whether or not the banker makes his character stand out as different from other characters.
Yes, I believe that the banker stands out as different from other characters in the story. Most of my evidence comes from the early parts of the story. Specifically, I think the banker distinguishes his character before the lawyer enters solitary confinement.
The first thing that I think is important for the banker's character is that he is the guy hosting the party. Wallflowers and people that are not confident and self-assured do not host parties. The banker has no problem hosting the party, nor does he have problems with hosting many distinguished guests. From the first paragraph, a reader already gets the feeling that the banker believes that he is an important man.
Then the banker talks, and my initial impression is that he has an elevated opinion about himself and his thoughts in relation to other people's thoughts. The first thing that he says about capital punishment is the following:
"I don't agree with you."
Notice the "you." The banker doesn't say "I don't agree with that." He directs his comment to an actual person. It's a confrontational way of entering the discussion. It's here that the reader might begin thinking that the banker is a bit of a pompous and arrogant bully, but there isn't enough evidence . . . yet.
Once the bet is made, the banker distinguishes his character by confirming earlier suspicions about his elevated opinion of himself and lack of respect for another person's opinion. The banker berates and teases the lawyer about the bet.
The banker, spoilt and frivolous, with millions beyond his reckoning, was delighted at the bet. At supper he made fun of the young man, and said: "Think better of it, young man, while there is still time. To me two millions are a trifle, but you are losing three or four of the best years of your life. I say three or four, because you won't stay longer. Don't forget either, you unhappy man, that voluntary confinement is a great deal harder to bear than compulsory. The thought that you have the right to step out in liberty at any moment will poison your whole existence in prison. I am sorry for you."
The banker is basically talking smack at a party for distinguished members of society. The guy might be a member of high class society by his wealth, but he for sure is low class by his actions. That character trait is further confirmed at the end of the story when readers discover that he is willing to commit murder in order to hold onto the rest of his wealth.