The click you're hearing as you plug your level 2 home charger into the car is simply a physical latch catching the tab on the standard j1772 'nozzle'. When you unplug it at home again you push the button on top of the J1772 'nozzle' disengaging the physical latch to remove it from the car.
At a public DCFC station when you plug the CCS 'nozzle' into the car and then activate a charge through that station's software (with a credit card account in a cell phone app or with an RFID card) the latch is activated via an electric solenoid device. You can cancel a charge and remove the 'nozzle' by pushing the button on top similar to your level 2 home charger but the difference is that the DCFC latch has a delayed electrically activated release.
Having a basket full of charging accounts from different providers to activate a charge can be cumbersome but some like 'Electrify America' just work with a credit card on the front of the DCFC unit. There's a learning curve and lots of different pricing structures involved for sure.
EVgo currently has a "Plug & Charge" setup you can sign up for that makes it so the BOLT just starts charging automatically when the CCS connector is engaged with the car. An EVgo charging account is still required.
https://cleantechnica.com/2022/09/22/ev ... utocharge/