Does the Ward keyword trigger Baral's card draw clause?
Yes, this is confirmed by the Strixhaven mechanics guide:

Ward is a triggered ability.

More generally, a triggered ability is defined as (Comprehensive Rules 603.1):

Triggered abilities have a trigger condition and an effect. They are written as “[When/Whenever/At] [trigger condition or event], [effect]. [Instructions (if any).]”

The reminder text for Ward meets that pattern; while reminder text formally has no bearing on the game, it is intended to be helpful and it would be very unhelpful for the reminder text to meet the pattern of a triggered ability without being one.