Nobody would say „trujący wąż” (poisonous snake) or „jadowity grzyb” (venomous mushroom). The distinction is similar to English. There are exceptions and contexts where it can be used interchangeably but arguably the same is true for English.
No? That's how I've always said it. "Ta żmija jest trująca" - don't see any issue here. Jadowity grzyb I'll agree.
You've never visited Italy ? They're not that far away and I'm sure you'll love it.