Interesting. I have two concerns with your "lime means Rose's lime juice" hypothesis.
(1) When I google "gin gimlet recipe", all search results I looked at (the top 4) call for "fresh lime juice" as the ingredient, leading me to believe that fresh lime juice is more common as a gimlet ingredient. The article you linked even states that fresh lime is more common: "A lot of bartenders out there adopt the fresh-lime-juice-and-sugar method of making lime cordial."
(2) Why would a recipe use the word "lime" to refer to Rose's lime juice? It seems like no one would ever be able to make the cocktail correctly if they had to infer that "lime" refers to Rose's lime juice rather than fresh lime juice.
(1) When I google "gin gimlet recipe", all search results I looked at (the top 4) call for "fresh lime juice" as the ingredient, leading me to believe that fresh lime juice is more common as a gimlet ingredient. The article you linked even states that fresh lime is more common: "A lot of bartenders out there adopt the fresh-lime-juice-and-sugar method of making lime cordial."
(2) Why would a recipe use the word "lime" to refer to Rose's lime juice? It seems like no one would ever be able to make the cocktail correctly if they had to infer that "lime" refers to Rose's lime juice rather than fresh lime juice.