What is the most accurate criterion for following a funeral procession to obtain the reward, and is it a condition to follow it from its departure from the house? And what are the most recommended stages for following it?

There is no contradiction between what Al-Shawkani mentioned and what came in Islamic jurisprudence regarding following a funeral. The full promised reward is only attained by whoever escorts the funeral to the cemetery until it is buried. Whoever only prays receives a qiraat less than whoever prayed and escorted. Al-Hafiz Ibn Hajar preferred the view that whoever only prays, without escorting, receives a qiraat, but it is less than the qiraat of one who escorted and prayed. It is not a condition to enter the house of the deceased's family to escort the funeral; rather, one can wait outside and follow it to the mosque for the prayer, and then to the cemetery to obtain the full reward. The righteous predecessors were keen on this reward, such as Ibn Umar, may Allah be pleased with him, who said, "We have missed out on many qiraats," after learning about the virtue of escorting a funeral until its burial.