What is the meaning of the phrases "peace be upon them" (عليهم السلام) and "may the prayers and peace of Allah be upon them" (عليهم الصلاة والسلام) for the Prophets, what is the difference between them, and which one is more firmly established by evidence?

It is legislated to invoke blessings upon all prophets, and the most complete form is to combine both blessings (Salah) and peace (Salam) upon them.

According to the majority of scholars, the meaning of invoking blessings (Salah) upon the Prophet, peace and blessings be upon him, is: from Allah, it is mercy; from the angels, it is seeking forgiveness; and from humans, it is supplication. Some meticulous scholars, such as Ibn Al-Qayyim and Ibn Uthaymeen, preferred the view that it means Allah's praise of him in the exalted assembly.

As for the meaning of invoking peace (Salam) upon him, peace and blessings be upon him: it is a supplication for his safety from every affliction during his life or after his death, and for the safety of his religion and his Sunnah.