I think the county is spelled Argyle, the parish, district & dukedom is Argyll.
I was and am under the impression that Argyle is simply an alternative, and much less common, way of spelling the name of the county.
Fullarton's Gazetteer (1842) says. "Argyle or Argyll, an extensive shire ....".
Putting "Argyle County Council" into a well-known search engine produces just 10 hits. "Argyll County Council" produces 1,210. "Argyllshire County Council" crops up 152 times, and "Argyleshire County Council" just once.
The GENUKI web site and Scotland's People both Use 'Argyll' for the county. So does this web site.
The NRS catalogue has 1755 items mentioning Argyle, only 44 of which are associated with the word 'County' and 21,724 mentioning Argyll, of which 2,617 are in association with the word 'County'.
So I will be sticking with using 'Argyll', as I always have done, to refer to the county, and using 'Argyle' for the the names of sundry streets, and of course for the well-known diamond pattern socks.