Does Lux still draw a card if you counter her spell?

No. Lux's legend ability only triggers when a spell is played (resolves), and a countered spell is not considered played (Rule 425.1.b, 419.4.b). Countering prevents resolution entirely, so Lux does not draw.

This is different from a spell whose targets become illegal — that spell still resolves (doing nothing), and Lux does trigger. But if it's countered (e.g., by Wind Wall), no draw.