So Router is what you get out of the box as you understood.
Repeater mode is then an extension of that. The router acts as a client, connecting to a remote wifi, while still acting as a router for your devices with firewall etc. Devices get their IPs from the GL router, not from the upstream network.
Extender and AP are modes where the firewall is disabled.
In Extender mode the router will connect to the remote wifi and then copy all the settings of that wifi (name, channel password etc) and extend that wifi using it’s radios. Devices get their IPs from the original network the router extended.
AP mode extends the network of the WAN port to a new wifi network. Devices get their IPs from the network on WAN.
Lastly there is a hidden feature, you can in Repeater mode turn the router into a wifi client only by turning off the wifi networks. This can be useful if you have for example a few LAN devices you want to connect to a wifi network, without creating your own secondary wifi network.