Your assumptions are right on the money. The scene was constructed to let these two heroes bond as well as communicate, displaying the similarities and the contrasts between them. Says the film's director John Woo in this interview:
First of all, our character [Zhou Yu], played by Tony Leung - in real history, he was a musician [see last paragraph here]. He knows so much about music, but at the meantime, he was always a great leader. And I took the idea of jazz. I have desgined the scene to let the two heroes, Zhou Yu and Zhuge Liang, Tony and Takeshi - they were using music to communicate each other. I think it made the scene more fun and I had never seen anything like it before in any of the Chinese movies.
Actor Takeshi Kaneshiro who plays Zhuge Liang elaborates about the musical duel:
The director said a high level skill wasn't that important. He wanted us to communicate the contrast of war and peace between the two heroes, and he wanted us to do it by playing the zither. The facial expression was what was important.
The scene is also a thematic and cinematic homage to the famous guitar and banjo 'duel' between two players from different walks of life in John Boorman's film Deliverance, where a suburban man and a rural boy bond over a musical duet. Similarly, Zhuge and Zhou 'duel' to relieve the tension between them, display their different ideologies and eventually understand each other and unite. From Kenneth E. Hall's book John Woo: The Films:
[Red Cliff] Editor David Wu confirmed that his intent was to recall the Boorman film. [...] Unlike the scene in 'Deliverance' Woo's scene is a case of warfare by other means. The latent rivalry between leaders Zhuge Liang and Zhou Yu must be resolved, and the musical duel is a harmless means of partially defusing it.
And I think I agree with your analysis of their styles. It makes sense that Zhou, the militant leader, will initially display a more aggressive tone while Zhuge, the clever and calm strategist who's hinted to be of humble farming origins, will prefer a more subtle, reflective tune. But I'd say that in the end, their styles more or less combine into a turbulent, rhytmic duet - telling us our two heroes must now join forces, and win the war together.