Why Sam was about to correct Dean about his “sitting shiva” comment?

You have got the essence of the joke, that they are not mourning a dead friend or relative, but are waiting for Shane/Prometheus to come back to life. Shivah is a somber occasion, with many cultural associations, most of which make no sense in the context Sam and Dean are in.

In Judaism, sitting shivah is a very solemn and important ritual. In fact, Judaism as a whole treats reverence for dead family members as one of the most important observances. Even many Jews who rarely attend synagogue may attend on the anniversary (Yiddish Yahrzeit) of a parent’s death, and the Mourner’s Kaddish is one of the oldest and most important Jewish prayers. (The Kaddish is one of the prayers that are essentially never translated from Hebrew, yet even many Jews who know no Hebrew know the Kaddish by heart.)

There are many ritual traditions associated with shivah—covering mirrors of the bereaveds’ home, sitting low to the ground, burning candles for seven days. These are not necessarily mandatory, but they play a cultural role in helping the living come to grips with the hard fact that someone they cared about is dead, making the transition from having the blessing of a living friend to having the blessing of their eternal memory. None of this makes sense, when the dead man is going to come back to life. There is no transition, no depth of mourning. So Dean’s description is totally inapt.