The point of GFCIs is to protect users from being electrocuted. They are generally required/recommended in sites around a residence where the probability of being shocked is highest, i.e. where the resistance to ground potential is lowest (usually because of the increased exposure to water) -- which would be bathrooms, kitchens, basements, and outdoors. A GFCI is designed to reduce the possibility of a lethal electrical current from passing through the heart or damaging other body parts.
Without saying more about how a GFCI operates, there is no question that 1) it is much safer to not use a plug-in EVSE, but to use a direct wired EVSE instead and 2) it is safer to use a plug-in EVSE to a GFCI receptacle than to a non-GFCI receptacle. Any built-in GFCI protection in an EVSE only protects past the control box (like at the J1772 connector), not at the plug. If a GFCI trips without clear reason, unless the GFCI is faulty there is usually current leakage somewhere in the wiring that needs to be corrected (probably by rewiring). Also you are unlikely to have nuisance trips if your EVSE is plugged into a dedicated circuit (preferably 20a btw because of the likely thicker gauge wire used is less likely to overheat).
Do not use extension cords for charging your EV, and especially not around water!
I am not a licensed electrician, so please feel free to correct me...