Just connecting a device to the internet doesn't leave it non-airgapped until it starts sending and receiving TCP or UDP traffic (collectively I will call this stuff TCP although there's a slight difference).

For example, if the device has any sort of incoming ports, then it is no longer air-gapped as hackers can start throwing malicious traffic at it.

Outgoing ports are trickier and depend on the server that said traffic is going to - if it is your own managed server from the internet, that's not making any other traffic, or running any other software on different incoming ports, using a protocol whose traffic you have audited, there shouldn't be any concern, however if it is some random website or even google.com or AWS, then the air-gapped nature is gone because you cannot verify what those servers are doing with your request.

You can have a LAN in an airgapped setting, even a WLAN - as long as all computers inside the network are under your control and don't talk to the outside world (with the possible exception of getting security updates from your package manager).