What is a Local RSS Client?
A local RSS client is an RSS client that you install on your PC as an application that fetches RSS feeds directly from the source. The term contrasts websites that let you create an account to access RSS feeds online via a web browser. This "RSS as a service" scheme may come with a subscription cost, while local RSS clients generally can be used for free since there is no middleman.
Sometimes the term desktop RSS client is used similarly to mean an RSS client for desktop, which you install, as opposed to a website. However, it's worth noting that nowadays it's common for websites to be packaged as desktop applications with technologies such as Electron.
Observations
Backwards: local RSS clients came first, so why do they have a qualifier instead of being the default? Shouldn't we be referring to online RSS clients with a qualifier instead? Perhaps this is because RSS as a service is more popular than local RSS clients!