In case if you don't know, Elasticsearch uses Lucene popular search engine library and you can consider Elasticsearch as distributed Lucene in action as well.
Elasticsearch distributed nature comes from the fact, that every ES index can be made of multiple shards(which internally a full-blown Lucene index) and these shards can be horizontally scaled on multiple physical nodes.
Don't confuse with naming convention and understand the concept as explained in the highlighted sentence of yours, that ES index is a logical grouping of all the shards(internally physical lucene shards), due to which its distributed nature comes in.
There are multiple resources available related to it and I just tried to summarize the important part of your question which is about shard(self-contained index).
Edit: : Recently Elasticsearch removed the types and reason of it as it uses Lucene explained in their official docs of removal types explaination which would help you understand better.