Is the intention behind washing the limbs of ablution in order during ritual bathing to perform the usual complete ablution?

What you mentioned, that a ghusl (definition) (ritual bath) for cleanliness or a supererogatory ghusl does not suffice for wudu' (ablution), is correct. This means that whoever performs a supererogatory or permissible ghusl, it does not suffice him for wudu' until he performs the prescribed wudu', fulfilling its obligations: intention, washing the four limbs, maintaining order, and continuity. If he performs this prescribed wudu' by having his wudu' limbs washed in order during his ghusl, it suffices him; otherwise, he must perform wudu' after the ghusl if he intends to pray.