How do we differentiate between the virtue of offering sincere acts of obedience to Allah Almighty and what the three men did when the rock fell upon them, as they supplicated through their righteous deeds to be relieved of their distress?

Working purely for the sake of Allah is better than working with the intention of drawing closer to Him and fulfilling a desire. The Hadith (definition) of the Cave does not contradict this, for the companions in the cave performed their deeds with sincere intentions for Allah, and then they supplicated through these deeds during their distress. Supplicating through righteous deeds is permissible and does not contradict sincerity; rather, it is a part of sincerity. Ibn Baz states that the permissible means (wasilah) is to seek closeness to Allah through faith and righteous deeds, which is what Allah mentions in His saying: {O you who have believed, fear Allah and seek the means [of nearness] to Him.} Performing a deed with the intention of seeking closeness and fulfilling a need does not negate sincerity, even though it is less virtuous than working purely for the sake of Allah.