John hi , the remote fobs you describe sound the same as my Mk3 ones , as for the keys if the non Lexus one is not a transponder key that has been programmed for your car then it will behave as you describe . I can only assume that pushing certain pedals overides the immobilizer in some way to allow the car to start ,if that is the case it is not much of a theft deterrent .
I have had a spare key done at a local key cutting shop, the wife looks after this in case I lock my keys in the boot again , the key opens all the doors and turns on the ignition but wont start the engine . Will have to try that pedal trick to see if I can start the engine with it .
Glad you have got your other fob working ,its good to have a spare .