What you refer to as logic AI is a subset of what is called symbolic AI, as you manipulate symbols, according to certain rules (which could be rules of logic). These rules are either authored by a human being, or they can be learned from examples. There are algorithms to derive decision trees (eg ID3) or other rule sets from training data. But the important aspect is that the 'intelligence' or 'knowledge' of the system is expressed in such rules and (usually specific) algorithms. Symbolic AI systems are generally designed for a particular purpose, and it can be difficult to use them for other tasks (eg language analysis vs. natural language generation)
Most learning AI nowadays operates on the sub-symbolic level: instead of manipulating symbols with a particular meaning (eg words in a human language) they operate on numbers. Symbols are mapped onto numbers (eg through embeddings, ie vectors of co-occurrence information). The algorithms used are general purpose: a neural network doesn't really care what it is used for, it simply learns to associate numerical input with numerical output. There is no human involvement apart from preparing and providing training data (and perhaps choosing a particular set of basic parameters/architecture), and the 'knowledge' is the model derived through training, basically a large blob of numerical values.
At the interface between the two parts is the mapping between symbols and their numerical representation.
As a diagram, imagine a two-way partition between symbolic and sub-symbolic AI, and a further set ('learning') which encompasses most of the sub-symbolic part and some (but not much) of the symbolic part.