You didn't back down you placated the family and that is different. It is called customer service in other businesses. You agreed to let the kid come in a pullup. Parents are paying for them so who cares - the training underwear is worn WITH a pullup or with plastic pants not alone as in no "accidents" at daycare on teh floors/furniture.

You agreed to "offer him the toilet" - you didn't commit to how often or in what capacity so you are still totally able to do it when and if and how you wish.

I always do a diaper at naptime for training kids anyways because we are not responsible for knowing we have to pee when we are sleeping so naptime accidents are ok - awake ones are not. We dont' pee in our pullups.