Bath, Shawnee and Perry high schools are not really in Lima, Ohio. They are referred to as Lima Bath, Lima Shawnee and Lima Perry but are not really in the city nor even its limits remotely. They all have Lima zip codes (in fact shawnee and perry share the same zip code some how). I have even seen Perry referred to as Cridersville Perry. Cridersville is a small town in southern Allen / northern Auglaize counties and has NO affiliation whatsoever with Perry schools. Kids who attend Cridersville Elementary go to Wapakoneta HS in fact.
Bishop Fenwick for years I've seen listed as Middletown Bishop Fenwick. Clearly they have a Franklin, OH address though.
Having played at Lima Senior back in the GMC days, I now realize that Sycamore, Princeton, etc... are not really in Cincinnati haha. Princeton does have a cincy address but Sycamore is listed in the town of Montgomery, OH.