Yes, they should be rated in BTUs per hour. Barbecue grill manufacturers aren’t really well known for their strict adherence to proper units.
It really is BTU/hr, but it’s such a widely used unit of thermal power that everybody just drops the /hr. It’s kind of like concrete being sold by the yard rather than by the cubic yard.
The square inch part refers to Heat Flux, or power per unit area. A acetylene torch puts out a lot of BTU/hr but is not a good method for bbq since it is too high on Heat Flux.