Whenever I think of WiFi, I think of connecting wirelessly to the network and to the internet, which is basically what a WLAN is, right?
I would suggest another way of thinking: WiFi or WLAN replace the wire with some kind of radio-based transfer in otherwise, well, cable-based networks. Here, the radio waves have the same function as the electricity in copper or the light in glass fibres.
This way of thinking puts your mind on the proper layer of the network stack, and makes it much easier to debug problems or plan proper solutions for your needs.
The physical layers are obviously completely different. But the similarities start to show up very early in the network stack, well below the IP/UDP layer one is familiar with. In fact, the engineers who have invented all of this have obviously taking great care that on top of the physical layers, things get done in an uniform, consistent manner as soon as possible.
If you get a WiFi device which allows or forces you to do very fine-grained custom setup (i.e. not the locked-down pre-configured standard WiFi "router" you get from 99% of the customer-oriented DSL providers these days), you'll notice that many aspects of your network setup do not care a bit about whether a given leg is cable- or radio-based.
The WiFi radio module in your router or smartphone has the same type of MAC address like the ethernet port in your laptop, and for example your separate DHCP server (which assigns IP addresses to MAC addresses) does not care (or usually even know) if the device lives on radio or the wire.
The packets being broadcast through air are a similar format (802.11 frames) as copper-based Ethernet frames and live on the same abstraction layer - i.e., they are transmitted between MAC addresses (i.e. the usually fixed, "burned-in" addresses of physical components in your devices/routers). If you have a generic enough WLAN device, you can easily treat it like just a way to forward 802.11 or even Ethernet packets over air, it does not need to know anything about IP package formats, routing, and so on (this would be useful for dumb WiFi expanders which just forward packages on a low level to avoid problems with distance or obstacles and which would not even need or have a MAC address of their own).
Neither has much to do with the "Internet" except insofar as with our everyday network technology, all these network technologies are very easy to connect to the Internet (and obviously are for 99% of the consumer WLANs). But that is not a property of WiFi/WLAN in itself, but happens on a higher layer.
Neither has much to do with WAN (wide-area-network) technology, aside, again, from being obviously more or less related in some aspects or on some of the abstraction layers.