Structurally you've got it right, and the labels do different jobs.
Room rental is the underlying arrangement. Each bedroom is leased separately, so the tenant has the exclusive right to one bedroom and shared use of the kitchen, living space, and bathrooms. That's the strict meaning and it describes both of your listings.
Co-living, used strictly, describes room rental that's been packaged as a managed service. The room comes furnished, utilities and internet are bundled into one monthly payment, someone cleans the common areas on a schedule, and there's often a single operator running many houses with consistent standards and sometimes community programming. Loosely, and this is how you'll usually see it in the wild, people slap co-living on any shared house listing because it markets better than rooms for rent. So the two houses you found may be identical in substance, with one of them charging more because the service layer is real and included.
Where the label stops mattering is in the local code. Cities and states define terms like rooming house, boarding house, and single room occupancy in their own ordinances, and those definitions carry the rules on licensing, occupancy caps, and inspections. Whether your building falls inside one of those definitions depends on the specific ordinance and how it's applied, which is a question for a local land use attorney. A website saying co-living carries no weight with a code officer holding a definition of family that caps unrelated adults at three.
One thing to know from the service angle since that's where you're looking: the bundled model shifts costs onto the operator that a normal landlord passes through, utilities, internet, cleaning, and consumables. Anyone pricing a co-living management service has to price those, not just the rent collection.
