There is one discussion on this topic here: https://www.geni.com/discussions/239585 - it focuses on whether Meyer's father David is the David who is son of Jacob and Toba (Fine) Rosenberg - and clearly shows the answer is No.
I think we are now clear about the Father - his given names were David Zev or Zeev and his surname was Herscovici.
His son Meyer and Meyer's children used the name Rosen in America.
There is a record in JewishGen - in Romania Vital Records Collection - Births - for the birth of Meyer's youngest child, Pauline, as
HERȘCOVICI, Pesa Ghitla -- born: 19/Feb/1896 - Father: Meier, 42 -- Mother: Freida Beila -- Comments: Declarant: Father. Born in: 6 Marta St. Father is a carpenter
So I think we can be sure Meyer and all his children were originally Herscovici.
Unless Meyer's father came to the America, I do not think he ever used the surname Rosen or Rosenberg.