Mining fulfils two roles:
- identity management: To identify who is a trustworthy block producer in an anonymous and trustless network, mining introduces a cost to the block generation process and we use a game theory assumption that it serves as a barrier for attackers. This doesn't mean there aren't validation checks performed by other nodes in the network and the consensus forming rules themselves of course.
- consensus formation: In a distributed network, nodes are bound to disagree on which is the canonical chain out of two or more competing forks. The most common algorithm calculates the accumulated computational effort of each chain of blocks and picks the "heaviest" one, ie the one with the most work done on it. This has the nice side effect of ensuring economic finality for the transactions in those blocks.
Now, regarding decentralisation you state that "the cryptocurrency people" use it wrong, but the definition you gave is incorrect. As a cryptocurrency person, Vitalik Buterin described decentralisation here: https://medium.com/@VitalikButerin/the-meaning-of-decentrali...