Your understanding is correct. Unincorporated Clark County and the City of Las Vegas are separate jurisdictions, and whether a property falls inside city limits or in the unincorporated county determines which office you file with, not both.
The short version of how it works in most Nevada markets: if an address is inside an incorporated city (Las Vegas, Henderson, North Las Vegas), that city's business licensing and short-term rental permit process applies. If the address is in unincorporated Clark County, meaning it has a county address but sits outside any incorporated city boundary, the county's process applies. The two systems do not overlap, so you would not typically need both. The confusion usually comes from addresses that look like they belong to a city but technically sit in unincorporated county land, which is genuinely common in the Las Vegas valley.
The one thing worth checking directly: Henderson has its own STR ordinance separate from Las Vegas city, so if that flip was in Henderson, the city of Henderson's licensing office is the right door, not Clark County and not the City of Las Vegas office.
I want to be straight with you: I am describing how the jurisdictional split generally works, and STR permit rules change often. Before filing anything, I would confirm the current requirements with the specific city or county office that covers that address. A call to Henderson's business licensing office or the Clark County Business License Department (depending on where the property sits) will get you a definitive answer faster than any forum will.
The strategy guide for short-term rentals on Rena covers regulatory risk in more detail under the guide tab if you want background on why this varies so much city to city.
Do you know whether the property address is technically inside Henderson city limits or in unincorporated county land? That single fact changes which office you need.
