I agree it can mean a lot of things. I would consider it being well rounded to some degree. Though, some of my hunting friends are amazing at one thing, like bow hunting white tails and are about as "accomplished" as one could be in that regard, but could find their backside with both hands when it came to waterfowling. So I would say someone could be an "accomplished deer hunter" or "elk hunter" or "big/small game hunter", or even "western hunter". There are very few people out there who could go to any continent and chase any game be it birds or ungulates or crocodiles and know the animals/habitat as well as an individual with years of experience and success in that setting.
I would then agree that most hunters accomplished in one area have a significantly shorter learning curve in another area similar to their area of expertise than a novice, but perhaps less so than an area far different from theirs. For example a accomplished deer hunter from the northeast will have an easier time adapting to mule deer or elk in Wyoming than an accomplished alligator hunter from LA. Yet neither are accomplished elk/mule deer hunters.
I will say though if you want to call yourself an accomplished hunter, you should know how to care for and break down game, not just gut it, throw it in the truck and take it to the butcher. Similar for equipment skills. An accomplished rifle hunter doesn't have cabelas mount their scope for them or have unkept gear. Again, my opinion.
I will say what has made me feel the most accomplished and like a woodsman is learning to trap. It has made my attention to detail in the woods and in animal behavior significantly magnified than prior to picking it up.
Interesting thread and I agree it's not heads on the wall. To me, it's who would you feel lucky to be in a survival situation with where woodsman ship and hunting would be important.