Must the Hatan’s Family Lineage Appear in the Ketuba?

When writing a ketuba, does the rabbi who fills in the ketuba need to write the lineage of the hatan, i.e., kohen, levi etc. Although the Shulhan Arukh does not mention this requirement, the Rama (Even Haezer 129:77) writes that kohen or levi should be added, lest there be more than one person with the same name. Furthermore, nowadays, one of the ways we verify whether a person is a kohen is to check the ketuba, and therefore it is proper to mention if the hatan is a kohen

If the hatan’s last name is Cohen, then it is unnecessary to write "ploni ben ploni hakohen lemishpahat kohen," rather, it is sufficient to write "ploni ben ploni lemishpahat kohen." R. Moshe Feinstein once ruled that if the ketuba identifies the hatan as a levi even if he is not a levi, the ketuba is still valid, as the hatan still accepted upon himself to pay the sum of the ketuba.