Even taking into account the difference in their characters, as you say, I don't think Lizzy could be expected to recognize that Darcy was courting her.

And from Darcy's point of view, I have a hard time believing that a man with his background really thought that behavior that fit the descriptions we have ("It seemed like wilful ill-nature, or a voluntary penance," "he frequently sat there ten minutes together without opening his lips; and when he did speak, it seemed the effect of necessity rather than of choice -- a sacrifice to propriety, not a pleasure to himself," etc.) constituted courting. I can't believe that if Darcy had been courting the daughter of the Earl of _________, he would have behaved that way.

Darcy may have thought that he had somehow communicated his intentions to Lizzy, perhaps inadvertantly, but I don't believe he thought he was really courting her.

I think Darcy's problem was that during that period, he was still experiencing the struggle he described in the proposal, and his behavior reflected that struggle. He wanted to marry Lizzy and at the same time, he couldn't bear to degrade himself. He knew he should stay away from her, but in spite of his efforts, he couldn't. He didn't want to give her any idea that he might marry her, since his reason rather than his emotion might still win out in the end, but he was still drawn to places he might meet her.