Traditionally a particular model derailleur with short, medium, long cage options only impacted the chain wrap. The max cog was the same because the derailleur body was identical with the cage being the only difference.
While I’m guessing things haven’t changed, but I’m really not sure about more current derailleur offerings since wider range cassettes have become more prevalent on road bikes.
Shimano docs can confirm if a particular model in short or long has the same max cog or if there are other design differences that increase max cog with a longer cage.
That said, I would be more inclined to find a compatible RD that offered both a larger max cog and wrap if contemplating lower cassette gearing.
John