I have been on a few organized trips, mostly to far-flung places. I've learned I like the idea of someone else handling all of the logistics for getting around a less-developed country. And if I'm traveling all that distance, I also like that the tour leaders know where the special birds are likely to be.
Where to go? I like to see new birds. I'd be lying if I said it wasn't about the list, but it's not ALL about the list. I'm also interested in exotic mammals, too. And I like to go to new countries and have new experiences. If I can combine them all, so much the better.
Comfortable lodging? Not the highest priority. I've stayed in thatched cabins with bats roosting in the thatch. I've slept in yurts where I took the mattress off the military cots and put it on the floor. My only requirement is some way to stay warm in cold climates, a hot shower every couple of days, and a mosquito net where the local fauna dictates.
As to whether it's a dedicated birding tour company as opposed to a general nature company, the answer is "dedicated birding, please." I've been on a trip with a lot of non-birding spouses. They don't know how to behave around the birds, and you wind up not seeing as much.
But if it's someplace that has significant cultural sights, I would hope that at least some time - maybe an afternoon or two - is dedicated to that, as long as the planned itinerary forewarns the group about it. For example, if you're in Samarkand, you need to spend some time visiting the Registan and some of the major tombs. But you don't need to visit every last sight in the Lonely Planet. And you better not be taken to carpet shops.
As to leaders, I would say there have been two leaders who spoiled a trip for me. One was just plain grumpy and didn't seem to like being with the group (I've met birders on other trips with that same company who had the same impression). The other wanted to be the star of the show and got noticeably upset when you tried to help the other clients get on a bird. That was his job. But I've yet to be on a trip where the leader didn't know his stuff.