The Bioshock games are linear in the sense of where you need to go in each level, but the way that you fight the enemies in the game is quite open-ended. In case you didn't notice, in the Bioshock games you use powers along with your normal weapons such as electricity, fire, freeze, tornade traps and many more and all throughout the games you have to collect a drug called adam and then spend it on upgrades to your abilities and weapons. You can't get every possible upgrade in one play-through of the game, so what you choose to focus on is up to you, which affects your play-style.
There are many ways to get through many situations in the Bioshock games, such as going in guns blazing and shooting everyone, sneaking around and using the environment to set traps and pick off your enemies one by one or hack into computers to take control of robots and stuff etc. There are many different ways to handle a situation in the Bioshock games. The games also have a karma system because to collect adam you have to find the little sisters carrying the big syringes and either kill and harvest them or help them, what you do will affect what happens later on, so I wouldn't call the games scripted.
Bioshock doesn't give you a big area where you can go wherever the hell you want like in Crysis, but the game is still quite open-ended and Bioshock Infinite will most likely follow the same thing. I mean in Bioshock Infinite it showed one of the powers to be controlling the crows, but there will probably be many more. That video would only show a very small part of the game.