Following the precedent set forth by the Sage Advice Compendium, "immunity to poison" includes both the Poisoned condition and the poison damage type
The Sage Advice Compendium has two instances of explaining what "poison" means:
Q. Does a monk’s Purity of Body feature grant immunity to poison damage, the poisoned condition, or both?
A. That feature grants immunity to both...
Q. Does the heroes’ feast spell grant immunity to poison damage or just the poison condition?
A. The heroes’ feast spell grants immunity to poison in any form—damage and the condition.
We note that both of these features use the same wording as Nature's Ward:
At 10th level, your mastery of the ki flowing through you makes you immune to disease and poison.
[...] The creature is cured of all diseases and poison, becomes immune to poison...
When you reach 10th level, you can't be charmed or frightened by elementals or fey, and you are immune to poison and disease.
Thus there is no reason to believe that Nature's Ward would work any differently. "Immunity to poison" means you are immune to both the Poisoned condition and poison-type damage.