"No experience [required]" and "[Experience] not required" mean what they sound like. Previous experience is not required. They don't mean that previous experience is forbidden or that you're automatically disqualified because you have previous experience.
What should I write in my resume in my previous work experience?
I'd suggest including your 1 year of experience at the coffee shop. It's relevant to the new positions that you're applying for.
The best-case scenario is that having the experience listed makes you look like a more desirable candidate than other applicants with no experience (you'll need less training, and 1 year is long enough to show that you're probably trustworthy and not a job hopper; not that job-hopping is much of a concern for these kind of roles).
The worst-case scenario is that they've stated 'no experience' because what they're really looking for are young candidates (i.e. under age 20) so that they can hire someone for less than the full minimum wage. In which case your experience won't help you (or hurt you), since the deciding factor will be whether or not you're over 20 years old.
So either listing your previous experience will benefit you, or at worst listing it will neither help nor hurt. That means the previous experience should be listed, in my opinion.