Oooooh, colloquial vs formal definitions. Everyone has that one word or phrase they're sick of people using incorrectly.
While it's true that Wifi is technically a standard for wireless electronic communication, society has come to understand it as a way to connect wirelessly to the internet. So if a coffeeshop advertises Free Wifi, but you show up and it's only LAN access, 95% of people will be disappointed (and the last 5% will laugh at the absurdity of a coffeeshop setting up a network for customers but not bothering to connect it to the internet).