I believe a large part of "contentiousness" is that understanding is not the central aim - that the motivating factor for one party (or both) is "winning the argument". It's a fine distinction to draw between that and "contending", since there obviously are times when we should defend the Church and each other against distortions and misrepresentations.
I have found that the biggest difference for me personally is maintaining a calmness in approach and a mindset that focuses on understanding instead of belittling. I also have found that anger and offense lead inexorably to contentiousness, so if I can avoid taking things personally I can "contend" (engage different viewpoints) without being contentious (attacking the person expressing the different viewpoint).
In other words, I can be assertive about my beliefs without belittling others with contrary beliefs - particularly if my primary focus is on understanding how the other person's beliefs actually might be able to help me modify or understand my own better. Once I say, "There's nothing I can learn from you," contentiousness is almost a foregone conclusion.
If the other person holds that same view, I can avoid contention only by being willing to walk away without "winning". I'm fine with that.