The work in the second camp, aimed at just getting systems to work, is usually called "weak AI" in that while we might be able to build systems that can behave like humans, the results will tell us nothing about how humans think. One of the prime examples of this is IBM's Deep Blue, a system that was a master chess player, but certainly did not play in the same way that humans do.
Somewhere in the middle of strong and weak AI is a third camp (the "in-between"): systems that are informed or inspired by human reasoning. This tends to be where most of the more powerful work is happening today. These systems use human reasoning as a guide, but they are not driven by the goal to perfectly model it.
A good example of this is IBM Watson. Watson builds up evidence for the answers it finds by looking at thousands of pieces of text that give it a level of confidence in its conclusion. It combines the ability to recognize patterns in text with the very different ability to weigh the evidence that matching those patterns provides. Its development was guided by the observation that people are able to come to conclusions without having hard and fast rules and can, instead, build up collections of evidence. Just like people, Watson is able to notice patterns in text that provide a little bit of evidence and then add all that evidence up to get to an answer.
Likewise, Google's work in Deep Learning has a similar feel in that it is inspired by the actual structure of the brain. Informed by the behavior of neurons, Deep Learning systems function by learning layers of representations for tasks such as image and speech recognition. Not exactly like the brain, but inspired by it.
The important takeaway here is that in order for a system to be considered AI, it doesn't have to work in the same way we do. It just needs to be smart.
Narrow AI vs. general AI
There is another distinction to be made here -- the difference between AI systems designed for specific tasks (often called "narrow AI") and those few systems that are designed for the ability to reason in general (referred to as "general AI"). People sometimes get confused by this distinction, and consequently, mistakenly interpret specific results in a specific area as somehow scoping across all of intelligent behavior.
Systems that can recommend things to you based on your past behavior will be different from systems that can learn to recognize images from examples, which will also be different from systems that can make decisions based on the syntheses of evidence. They may all be examples of narrow AI in practice, but may not be generalizable to address all of the issues that an intelligent machine will have to deal with on its own. For example, I may not want the system that is brilliant at figuring out where the nearest gas station is to also perform my medical diagnostics.
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