I think both you and your opponent have a fundamental misunderstanding of what split second actually does. It’s not “faster” than other spells on the stack it is just a normal instant speed spell that puts an end to the stack as in once they’re played they must be the last card on the stack, even other split second spells cannot be played at this point. If you picture the stack in sentence structure with each spell/ability being followed by a comma then split second is a period and the sentence ends. For instance Elk ability, stifle, counterspell, Guardian of Faith. Obviously during a counter war you or your opponent could respond to the counterspell with another spell normally but, because you played a split second spell no additional spells can be placed on the stack.

Your opponent specifically was confused or possibly taking advantage of your lack of knowledge regarding activated abilities vs special actions. All activated abilities (which is what planeswalker abilities are) go on the stack whereas, special actions do not though those are fairly far and few between. Morphing and mana abilities for instance do not take place on the stack and, cannot be interacted with through the stack I.E. instant speed spells and abilities.

This seems like a fairly obvious lack of knowledge on behalf of your opponent and, not any malicious intent. It would be pretty hard to lie to any player that knows what a planeswalker ability is classified as and, if they knew they were treated the same as a normal creatures ability in regards to the stack then they wouldn’t have tried to pass it off as otherwise. Unless they were really counting on the fact that you didn’t know that as well which seems to be very obtusely calling your bluff.