I should have said "the factory provided portable trickle charging station". I suppose a car dealer could give you something else. There is no splitter. Even if a charging station can take either 120V or 240 V, it is only one type of voltage at a time, and no other voltages can be accepted other than the one you are using.
Electric code says you can't put a device on a circuit that will pull more than 80% of circuit rated amps. Therefore, a 30 amp circuit can only be used with a 24 amp or less EVSE. A 40 amp circuit can be used with up to a 32 amp EVSE, etc.
Yes, either a portable or wall mounted 240 V EVSE is probably best. You can mount it on the wall if you like or let it dangle from the receptacle, but some are so large and heavy that they can not dangle and must be wall mounted. You need to look at the models you are considering. Also for safety, I recommend you stay away from an EVSE that is not Intertek or UL approved.
You are right, the car either has a 3.6 kW (15 amp at 240V AC) or 7.2 kW (30 amp at 240 V AC) on board charger, so if you like, get an EVSE that can charge at the max amps the car can take. Or you can buy one that can deliver more amps (for a future EV in your future that can take more amps). Hope this helps.