We could classify AI in an infinite number of other ways; some based on outcome, others based on how it works on a technical level. Here are some examples:
Strong and Weak AI
Your first 'School' of AI is probably Artificial general intelligence "(AGI) is the intelligence of a machine that could successfully perform any intellectual task that a human being can. It is a primary goal of some artificial intelligence research and a common topic in science fiction and future studies." This is also known as Strong AI.
Your second 'school' of AI is known as a 'Expert Systems' which are defined as a system designed to emulate the decision making of a human expert in a specific domain. For instance, Deep Blue, the famous chess 'AI' is an Expert System in the domain of Chess; it emulates the decisions of human Chess experts. this is also known as Weak AI, and Narrow AI.
In this system, we are defining AI based on whether or not it possesses human-like cognitive abilities.
You could classify AI into groups of those that can pass the Turing Test and those that cannot. One isn't necessarily better than the other; i.e. just because something does not communicate to humans doesn't mean it isn't intelligent.
What 'Type' of 'Mind' it has
You can classify AI into what I would describe as what 'type' of 'mind' it has:
- Type I : Purely Reactive
- Type II : Limited Memory
- Type III: 'Theory of Mind'
- Type IV : Self-aware
How they were made
You could classify AI into those that were developed 'by man' and those that were developed using evolutionary algorithms and/or machine learning. Deep blue, was programmed with a defined set of rules, whereas the Google AI 'DeepMind' that can win at 'Go' is much more of an evolutionary algorithm, and effectively taught itself to win.